Stieltjes Functions: Theory & Applications

• Stieltjes functions are defined by a Cauchy-type integral representation using a positive measure, ensuring complete monotonicity and holomorphic extension to a slit complex plane.
• Generalized classes Sₐ introduce an order parameter and differential conditions that expand classical definitions and connect to Laplace transform representations.
• Their applications span special functions, moment problems, and matrix-valued analyses, providing insights into zero-free regions, stability, and analytic uniqueness.

Stieltjes functions, in the classical sense, are functions on $(0,\infty)$ or on a slit complex plane that admit a Cauchy-type integral representation against a positive measure, typically

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$

with the integral convergent for $x>0$. They form a distinguished cone at the intersection of complete monotonicity, Laplace transform theory, Pick–Nevanlinna theory, moment problems, Bernstein functions, and special-function analysis. Modern developments enlarge this picture in several directions: generalized Stieltjes classes $S_\alpha$ of arbitrary order $\alpha>0$, differential and finite-order characterizations, exact-order invariants, matrix-valued analogues, and applications to hypergeometric, gamma, and Lambert $W$ functions, as well as to ratios of entire functions and moment problems (Karp et al., 2011, Koumandos et al., 2017, Fritzsche et al., 2015).

1. Classical definition and analytic characterizations

A classical Stieltjes function is usually defined by the representation

$f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$

or, in equivalent real-variable form,

$$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$

where the constant is nonnegative and the representing measure is positive (Koumandos et al., 2017, Karp et al., 2017). In the generalized theory developed by Sokal, Berg, and others, the classical class is denoted $S_1$, emphasizing that it is the order-$1$ case of a larger hierarchy (Karp et al., 2011).

Several complex-analytic characterizations are standard. One form states that a function $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$0 is Stieltjes if and only if it extends holomorphically to $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$1, is nonnegative on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$2, and satisfies

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$3

(Kalugin et al., 2011). Closely related slit-plane classes $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$4 and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$5 are defined by the sign of the imaginary part in the upper half-plane together with positivity on the positive axis; they admit integral representations with kernels $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$6 and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$7, respectively (Katsnelson, 2011).

The classical class is tightly linked to complete monotonicity. A function $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$8 is completely monotone on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$9 if

$x>0$0

By Bernstein’s theorem, this is equivalent to $x>0$1 being a Laplace transform of a positive measure (Koumandos et al., 2017). Every Stieltjes function is completely monotone, and Widder’s real-variable characterization identifies exactly which completely monotone functions are Stieltjes: $x>0$2 is Stieltjes if and only if $x>0$3 is completely monotone for every $x>0$4 (Koumandos et al., 2017). This criterion remains one of the central structural tools in the subject.

2. Generalized classes $x>0$5 and exact Stieltjes order

For $x>0$6, a generalized Stieltjes function of order $x>0$7 is a function of the form

$x>0$8

with $x>0$9 positive and the integral convergent; the corresponding class is denoted $S_\alpha$0 (Berg et al., 2019). The same class admits a Laplace-transform representation

$S_\alpha$1

where $S_\alpha$2 is completely monotone (Berg et al., 2019). This makes explicit that generalized Stieltjes functions are Laplace transforms of $S_\alpha$3 multiplied by a completely monotone kernel. A basic inclusion holds: $S_\alpha$4 (Karp et al., 2011).

The order parameter is not merely formal. The paper "Generalized Stieltjes transforms: basic aspects" defines the exact Stieltjes order

$S_\alpha$5

proves that this infimum is attained, and gives a criterion for exactness in terms of monotonicity of a fractional integral $S_\alpha$6 built from a representing measure at a larger order (Karp et al., 2011). Compact support of the representing measure forces exactness, and hypergeometric examples show that nontrivial special functions can have explicitly computable exact order (Karp et al., 2011).

Within the generalized hierarchy, the class $S_\alpha$7 is distinguished. A function $S_\alpha$8 is logarithmically completely monotone if $S_\alpha$9 is completely monotone. Horn’s theorem characterizes this by complete monotonicity of all positive powers $\alpha>0$0. A central result used repeatedly in special-function applications is

$\alpha>0$1

where $\alpha>0$2 denotes the class of logarithmically completely monotone functions (Berg et al., 2019). The same source stresses that this inclusion is special to order $\alpha>0$3: for $\alpha>0$4, there exist functions in $\alpha>0$5 (Berg et al., 2019). This distinguishes order $\alpha>0$6 from the rest of the generalized scale.

3. Differential tests, finite-order truncations, and real-variable structure

The generalized order-$\alpha>0$7 theory has a precise real-variable characterization. Sokal introduced differential operators

$\alpha>0$8

and proved that, for $\alpha>0$9, the condition

$W$0

is equivalent to $W$1 being a generalized Stieltjes function of order $W$2 (Koumandos et al., 2017). Koumandos and Pedersen simplified this by introducing

$W$3

and proving

$W$4

Hence $W$5 is generalized Stieltjes of order $W$6 if and only if every $W$7 is completely monotone (Koumandos et al., 2017). This reformulation compresses Sokal’s two-parameter inequality array into one family of complete-monotonicity conditions.

The same paper studies finite truncations. For fixed $W$8,

$W$9

and $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$0 is exactly the generalized Stieltjes class of order $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$1 (Koumandos et al., 2017). Membership in $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$2 is characterized by positivity properties of derived measures

$f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$3

in the distribution sense, where $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$4 is the representing measure in a Laplace form

$f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$5

(Koumandos et al., 2017). This finite-order theory shows that requiring only finitely many complete-monotonicity conditions imposes a truncated hierarchy of measure-theoretic positivity constraints rather than full generalized Stieltjes structure.

A different finite-order program appears in "Stieltjes functions of finite order and hyperbolic monotonicity" (Bondesson et al., 2016). There, $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$6 denotes the class of nonnegative functions satisfying Widder-type inequalities only up to order $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$7, interpreted in the measure sense: $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$8 These functions need not be smooth. For $f(z)=a+\int_0^\infty \frac{d\mu(t)}{z+t},$9, they admit a unique representation

$$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$0

where $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$1 is a nonnegative truncated Laurent kernel converging pointwise to $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$2 as $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$3 (Bondesson et al., 2016). Two finite-order notions therefore coexist in the literature: truncated $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$4-complete-monotonicity in the generalized $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$5 framework, and truncated Widder inequalities in the classical $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$6-framework.

4. Special functions, transcendental equations, and zero-free phenomena

A large modern literature identifies explicit special functions as members of Stieltjes classes. For generalized hypergeometric functions, several parameter regimes imply that $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$7 is a Stieltjes function, while Meijer $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$8-based Laplace representations yield complete monotonicity for related $$f(x)=c+\int_{[0,\infty)}\frac{1}{x+t}\,p(dt),\qquad x>0,$$9 and $S_1$0 families (Karp et al., 2017). These representations are then used to compute branch-cut jumps and average values, obtain generalized Stieltjes formulas with kernels $S_1$1, and derive univalence and distortion inequalities in the half-plane $S_1$2 (Karp et al., 2017).

The order-$S_1$3 class is especially rich in explicit examples. Nielsen’s beta function

$S_1$4

is shown to belong to $S_1$5, hence to $S_1$6, and this is extended to broad alternating series families, ratios of Gamma functions, Prym’s function, and remainders in asymptotic expansions of Barnes’ double Gamma function (Berg et al., 2019). The same source proves, for instance, that

$S_1$7

for $S_1$8, while

$S_1$9

for $1$0 and $1$1 (Berg et al., 2019). These results connect Stieltjes structure to logarithmic complete monotonicity and to infinite divisibility of associated measures.

Lambert $1$2 furnishes a useful counterpoint. The paper "Stieltjes, Poisson and other integral representations for functions of Lambert $1$3" proves that

$1$4

is a Stieltjes function, and derives explicit Stieltjes integrals for $1$5, $1$6, and $1$7 (Kalugin et al., 2011). By contrast, $1$8 itself is not a Stieltjes function in the strict sense of the classical definition; rather, it belongs to the Bernstein and complete Bernstein frameworks (Kalugin et al., 2011). This distinction is a common source of confusion because the same special function may generate both Stieltjes and non-Stieltjes combinations.

The Stieltjes cone also yields sharp uniqueness and zero-free statements. For equations of the form

$1$9

with $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$00, the paper "Some transcendental equations on the Stieltjes cone" shows that there is no solution or at most one solution in the cut plane, and that any solution is real, positive, and bounded in terms of a critical value $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$01 (Giraldi, 2015). Applied to special-function Stieltjes representations, this produces zero-free regions for incomplete gamma, exponential integral, complementary error, Whittaker, Bessel, Lambert $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$02, and Binet functions (Giraldi, 2015). This suggests a general mechanism: once a special function is recognized as a Stieltjes transform of a nonnegative kernel, strong geometric information in the slit plane often follows with little additional input.

5. Entire functions, matrix-valued analogues, and moment problems

Stieltjes structure extends naturally to ratios of entire functions. Let $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$03 be a real entire function of genus $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$04 with only non-positive zeros and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$05 for $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$06. For shift arrays $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$07 and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$08, the ratio

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$09

has logarithmic derivatives that become generalized Stieltjes functions under weak supermajorization and power-sum cancellation conditions (Askitis et al., 2021). In particular,

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$10

is generalized Stieltjes of order $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$11, and under $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$12,

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$13

is generalized Stieltjes of order $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$14 (Askitis et al., 2021). Applications include Euler’s gamma function, Barnes’ $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$15-function, and multiple gamma functions, as well as an explicit link to Prouhet–Tarry–Escott identities through the same power-sum constraints (Askitis et al., 2021).

A different construction connects Stieltjes classes to zero localization of entire functions. If $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$16 belongs to the slit-plane Stieltjes class $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$17, then

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$18

is a Hurwitz stable entire function when $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$19 is generic; similarly, if $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$20, then

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$21

is Hurwitz stable unless $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$22, in which case $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$23 is Hurwitz stable (Katsnelson, 2011). The same paper extends the construction by replacing $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$24 with an entire function from the Laguerre–Pólya class of type I, thereby generating broad families of Hurwitz stable entire functions from Stieltjes data (Katsnelson, 2011).

Matrix-valued Stieltjes functions furnish the natural noncommutative generalization. For a real $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$25, the class $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$26 consists of $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$27 matrix-valued holomorphic functions on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$28 with positive semidefinite imaginary part on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$29 and positive semidefinite values on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$30 (Fritzsche et al., 2015). These functions admit integral representations such as

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$31

and, for a distinguished subclass,

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$32

with non-negative Hermitian matrix measures (Fritzsche et al., 2015). The framework is closely tied to truncated matricial Stieltjes moment problems, and it is stable under a Moore–Penrose inverse transformation: $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$33 whenever $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$34 (Fritzsche et al., 2015). Range and kernel of $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$35 are determined entirely by the measure parameters, which is crucial for matricial moment theory (Fritzsche et al., 2015).

6. Finite-horizon, differential-calculus, and alternative usages of the term

The term “Stieltjes functions” also appears in a distinct real-variable literature on Stieltjes differential calculus. There the central datum is a derivator $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$36, usually nondecreasing and left-continuous, which induces a Stieltjes measure $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$37, a $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$38-topology, a Stieltjes derivative $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$39, and function spaces such as $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$40, $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$41, $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$42, and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$43 (Fernández et al., 2022, Fernández et al., 31 Mar 2025). In this setting, the phrase “Stieltjes function spaces” refers to spaces of functions defined or differentiated with respect to $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$44, not to the classical holomorphic Stieltjes-transform class $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$45 (Fernández et al., 2022).

This alternative usage has developed its own compactness and approximation theory. Compactness criteria of Ascoli–Arzelà and Kolmogorov–Riesz type are proved for $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$46, $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$47, and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$48; Stieltjes–Sobolev spaces $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$49 are shown to be Banach, reflexive for $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$50, and compactly embedded into $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$51 or $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$52 under appropriate hypotheses (Fernández et al., 2022). The paper "On the approximation properties of Stieltjes polynomials" introduces $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$53-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function, and proves that when the derivator $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$54 has finitely many discontinuities, the space of $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$55-polynomials is dense in the space of uniformly $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$56-continuous functions (Cora et al., 7 Jul 2025).

Recent work has also highlighted structural differences from classical differential calculus. The kernel of the Stieltjes derivative can be nontrivial, so first-order Stieltjes differential equations may fail to have unique solutions in natural $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$57-differentiable classes; this motivates the introduction of spaces such as $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$58 and refined metrics adapted to $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$59-derivatives (Fernández et al., 31 Mar 2025). An additional generalization allows derivators of bounded variation that are not monotone, introducing functions of controlled variation and $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$60-exponential maps in the signed-measure setting (Frigon et al., 11 Jan 2025). These developments belong to Stieltjes differential systems rather than to classical Stieltjes transforms, and the distinction is terminologically important.

The coexistence of these usages can obscure the literature. In one tradition, a Stieltjes function is a Pick–Nevanlinna-type transform of a positive measure on $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$61. In the other, “Stieltjes” modifies derivative, integral, topology, or Sobolev space relative to a derivator $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$62. The two are linked by measure-theoretic ancestry but are not interchangeable notions (Fernández et al., 2022, Fernández et al., 31 Mar 2025).

7. Conceptual role and enduring themes

Across its variants, the theory is organized by a small number of persistent principles. First, positivity of the representing measure governs analyticity, monotonicity, and slit-plane geometry. Second, generalized orders $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$63 and finite-order truncations quantify how strongly a function belongs to the Stieltjes world. Third, differential criteria convert integral representations into verifiable real-variable inequalities. Fourth, special functions often reveal hidden Stieltjes structure only after parameter restrictions, logarithmic differentiation, or algebraic modification. Fifth, matrix-valued and entire-function generalizations show that the theory is not confined to scalar transforms but interacts naturally with operator theory, zero distribution, and moment problems (Karp et al., 2011, Koumandos et al., 2017, Askitis et al., 2021, Fritzsche et al., 2015).

Two misconceptions recur. One is that every completely monotone function is Stieltjes; the classical theory distinguishes the Stieltjes cone as a proper subclass characterized by additional differential or complex-analytic constraints (Koumandos et al., 2017, Kalugin et al., 2011). The other is that “generalized Stieltjes” simply means “higher order and therefore stronger”; in fact the classes are nested upward in the order parameter,

$$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$64

so larger $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$65 gives a larger class, while exact order isolates the minimal admissible exponent (Karp et al., 2011). A related caution is that order $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$66 has exceptional logarithmic complete monotonicity properties that do not persist for all $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$67 (Berg et al., 2019).

The contemporary literature therefore presents Stieltjes functions not as a single isolated definition, but as a family of interlocking frameworks. The classical class $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$68 remains the prototype; generalized classes $$f(x)=c+\int_{0}^{\infty}\frac{d\mu(t)}{x+t}, \qquad c\ge 0,$$69 organize scale and inclusion; finite-order theories capture truncated positivity; special-function applications exhibit concrete analytic power; matrix and entire-function extensions connect the subject to moment problems and stability; and Stieltjes differential calculus supplies a separate, measure-driven real-variable branch. Together these strands explain why Stieltjes functions continue to occupy a central position in modern analysis.