Turán Inequalities: Special Functions & Polynomials

• Turán inequalities are a class of mathematical relations that establish log-concavity and real-rootedness criteria in sequences and special functions, notably in orthogonal polynomial systems.
• They employ analytic, differential, and integral techniques to derive sharp bounds, as illustrated by the behavior of modified Bessel functions and sequences like partition functions.
• These inequalities underpin stability analyses in spectral theory and combinatorial asymptotics, while extending to higher order forms and Jensen polynomials with significant applications.

Turán inequalities are a comprehensive class of inequalities centered on quadratic or higher-order relations among terms in sequences or values of special functions, most famously in orthogonal polynomial systems, special function theory, combinatorics, and analytic number theory. At their core, Turán inequalities provide quantitative log-concavity or real-rootedness criteria and admit powerful applications to spectral analysis, combinatorial asymptotics, and the theory of entire functions.

1. Classical Turán Inequalities: The Prototype and Basic Definitions

The archetype of a Turán inequality is the quadratic form

$T[f;x] = f(x)^2 - f(x-1)\,f(x+1),$

where $f(x)$ is a function (often a classical orthogonal polynomial, or a sequence of combinatorial numbers). For the modified Bessel functions,

$$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$

with $x>0$ and $\nu\in\mathbb{R}$, are prototypical Turánian expressions for special functions (Baricz, 2012).

For a real sequence $\{a_n\}$, the second-order (classical) Turán inequality is

$a_n^2 - a_{n-1}a_{n+1} \geq 0$

for all $n\geq1$, equivalent to log-concavity. Log-concavity and related inequalities figure prominently in the theory of real-rooted polynomials, Newton's inequalities, and the characterization of the Laguerre–Pólya class (Chen et al., 2017).

2. Sharp Turán-Type Bounds for Bessel and Special Functions

Turán inequalities for modified Bessel functions, both first ($I_\nu$) and second ($K_\nu$) kind, admit sharp two-sided bounds for their Turánians. For $f(x)$0 ($f(x)$1): $f(x)$2 with

$f(x)$3

and both bounds are asymptotically sharp as $f(x)$4 (Baricz, 2012).

For $f(x)$5 ($f(x)$6),

$f(x)$7

with

$f(x)$8

The bounds reverse for $f(x)$9.

These sharpen and generalize the classical relations $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$0 and $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$1, surgically refining dependence on the order and argument (Baricz, 2012).

3. Methodologies: Analytic, Differential, and Integral Techniques

The proofs utilize tools from ODE theory, functional analysis, and analytic number theory:

• Gronwall’s Differential Approach: By analyzing the logarithmic derivative $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$2, which satisfies a Riccati equation, one bounds $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$3 via estimates on $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$4 and connects the Turánian to differential inequalities (Baricz, 2012).
• Ismail’s Integral Formula: For $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$5, an integral representation for the quotient involving Bessel functions of the first and second kind is used to derive sharp monotonicity and bound properties.
• Hartman–Watson Integrals: Refined by classical integral identities, such as Nicholson’s formula, to achieve tighter inequalities. Similar analytic strategies extend to Struve functions (Baricz et al., 2014), Coulomb wave functions (Baricz, 2015), and confluent hypergeometric functions (Baricz et al., 2014, Baricz et al., 2011).

4. Higher Order Turán Inequalities and Jensen Polynomials

The scope of Turán inequalities expands to higher order forms, providing powerful conditions for hyperbolicity of Jensen polynomials: $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$6 and the equivalent statement that all roots of the associated degree-$$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$7 Jensen polynomial $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$8 are real (Ono et al., 2022, Craig et al., 2019, Chen et al., 2017). For the partition function $$T[I_\nu;x] = I_\nu(x)^2 - I_{\nu-1}(x)\,I_{\nu+1}(x), \quad T[K_\nu;x] = K_\nu(x)^2 - K_{\nu-1}(x)\,K_{\nu+1}(x),$$9 and $x>0$0-regular partition function $x>0$1, this translates to log-concavity, cubic, and general degree-d Turán inequalities for sufficiently large $x>0$2. For example, log-concavity (the $x>0$3 case) is equivalent to $x>0$4 (Chen et al., 2017, Craig et al., 2019).

Asymptotic formulas (e.g., using the Hardy–Ramanujan–Rademacher expansion) validate that for large $x>0$5, functions like $x>0$6 and plane partition counts $x>0$7 satisfy all higher Turán inequalities: all corresponding Jensen polynomials become hyperbolic, an analytic reflection of the underlying modularity and distribution of zeros (Ono et al., 2022, Banerjee et al., 21 May 2025, Dong et al., 2022).

5. Turán Inequalities for Orthogonal Polynomials and Generalizations

Orthogonal polynomials exhibit canonical Turán-type behavior. For Legendre polynomials $x>0$8,

$x>0$9

with strict positivity for $\nu\in\mathbb{R}$0 (Kahler, 6 May 2026). For Jacobi polynomials $\nu\in\mathbb{R}$1, Turán’s inequality holds on $\nu\in\mathbb{R}$2 if and only if $\nu\in\mathbb{R}$3 (Kahler, 6 May 2026). Two general recurrence-based criteria provide sufficient conditions for Turán positivity, applicable far beyond classical cases, and allow explicit nonnegative sum-of-squares decompositions of the Turánians for entire families of orthogonal polynomials, including sieved and generalized Chebyshev systems. These criteria have recently been detailed and extended (Kahler, 6 May 2026, Krasikov, 2011, Krasikov, 2023).

6. Applications and Extensions: Special Functions, Partition Theory, and Convex Geometry

• Special Function Theory: Turán inequalities exist for Struve functions, Coulomb wave functions, Krätzel functions, and generalized inverse trigonometric/hyperbolic functions, including refined bounds on functional ratios, complete monotonicity properties, and log-convexity in parameters (Baricz et al., 2014, Baricz, 2015, Baricz et al., 2011, Baricz et al., 2013).
• Partition-Theoretic Sequences: For $\nu\in\mathbb{R}$4, $\nu\in\mathbb{R}$5, plane partitions $\nu\in\mathbb{R}$6, broken $\nu\in\mathbb{R}$7-diamond partitions $\nu\in\mathbb{R}$8, and unimodal sequence counts, higher-order Turán inequalities are established via detailed asymptotic and probabilistic arguments (Chen et al., 2017, Craig et al., 2019, Ono et al., 2022, Banerjee et al., 21 May 2025, Dong et al., 2022).
• Convex Geometry and Polynomial Oscillation: For algebraic polynomials with zeros in convex planar sets, Turán-type oscillation (inverse Markov) inequalities relate the $\nu\in\mathbb{R}$9 boundary norm of the derivative to the function's own norm. In convex polygons and disks, the best possible lower bound is order $\{a_n\}$0 in degree (Glazyrina et al., 2024).

7. Impact, Broader Context, and Open Directions

The theory of Turán inequalities underpins stability criteria in PDE spectral theory, determines zero spacing in orthogonal polynomial systems, and quantifies real-rootedness and log-concavity in combinatorics. Recent advances have interpreted extremal cases via potential theory and energy minimization—most notably, the sharp constant $\{a_n\}$1 in the Erdős–Turán discrepancy for root distribution via equilibrium measures and logarithmic interactions on the circle (Shu et al., 2021).

The general methodology—leveraging ODE, recurrence, integral representations, and probabilistic tools—extends to a wide variety of analytic and combinatorial structures. Current research explores higher-order inequalities, connections with the Riemann Hypothesis (via Jensen polynomials of the Xi-function), and applications to stochastic process theory and statistical mechanics (Chen et al., 2017, Ono et al., 2022, Kahler, 6 May 2026).

Open questions include the determination of optimal constants in strengthened Turán inequalities (such as for ultraspherical and Hermite polynomials (Krasikov, 2023)), the extension to further families of special and $\{a_n\}$2-orthogonal polynomials, and the detailed classification of log-concavity and higher Turán positivity thresholds for partition-type and unimodal sequences in all parameter regimes.