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Hermite-Hadamard type inequalities by using Newton-Cotes quadrature formulas

Published 5 Dec 2023 in math.GM | (2401.08620v1)

Abstract: A convex function $f:[a,b]\to\mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a{b}f(t)dt\leq \frac{f(a)+f(b)}{2}. $$ Motivated by the above estimates, in this paper we consider approximately monotone and convex functions, and give upper and lower bounds to the numerical integral mean, i.e., to $\frac1{b-a}\mathcal{I}{n}(f)$, where $\mathcal{I}{n}(f)$ denotes some of the most popular Newton-Cotes quadrature formulas.

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References (11)
  1. Goswami, A.R., Páles, Z.: On approximately monotone and approximately Hölder functions. Period. Math. Hungar. 81(1), 65–87 (2020) Goswami and Páles [2021] Goswami, A.R., Páles, Z.: Characterization of approximately monotone and approximately Hölder functions. Math. Inequal. Appl. 24(1), 247–264 (2021) Goswami and Páles [2023] Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Goswami, A.R., Páles, Z.: Characterization of approximately monotone and approximately Hölder functions. Math. Inequal. Appl. 24(1), 247–264 (2021) Goswami and Páles [2023] Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  2. Goswami, A.R., Páles, Z.: Characterization of approximately monotone and approximately Hölder functions. Math. Inequal. Appl. 24(1), 247–264 (2021) Goswami and Páles [2023] Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  3. Goswami, A.R., Páles, Z.: On approximately convex and affine functions. J. Math. Inequal. 17(2), 459–480 (2023) Páles [2003] Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  4. Páles, Z.: On approximately convex functions. Proc. Amer. Math. Soc. 131(1), 243–252 (2003) Makó and Páles [2012] Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  5. Makó, J., Páles, Z.: On ϕitalic-ϕ\phiitalic_ϕ-convexity. Publ. Math. Debrecen 80(1-2), 107–126 (2012) Burden and Faires [2011] Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  6. Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Cengage Learning, Boston (2011) Hadamard [1893] Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  7. Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 9, 171–215 (1893) Hermite [1883] Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  8. Hermite, C.: Sur deux limites d’une intégrale définie. Mathesis 3(1), 1–82 (1883) Bessenyei and Páles [2003] Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  9. Bessenyei, M., Páles, Z.: Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6(3), 379–392 (2003) Bessenyei and Páles [2006] Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  10. Bessenyei, M., Páles, Z.: Characterizations of convexity via Hadamard’s inequality. Math. Inequal. Appl. 9(1), 53–62 (2006) Dragomir and Pearce [2003] Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003) Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)
  11. Dragomir, S.S., Pearce, C.: Selected topics on Hermite-Hadamard inequalities and applications. Science Direct Working Paper No S1574-0358(04)70845-X (2003)

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