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Exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions

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Pith's one-line read The paper establishes explicit parameter inequalities under which normalized confluent hypergeometric, Lommel, and generalized Struve functions belong to the exponential convex class $K_e$ or the exponential starlike class $S_e^*$, by…

desk verdict The paper is a competent extension of the authors' own exponential-starlike/convex program to three more special function families, but Theorem 2.3 as printed states the wrong function and must be corrected. read the letter →

arxiv 1908.07266 v1 pith:ZMPIDYVS submitted 2019-08-20 math.CV

Adiba Naz , Sumit Nagpal , V. Ravichandran This is my paper · ORCID
classification math.CV MSC 30C1030C4530C80
keywords differentialsubordinationexponentialfunctionconvexstarlikeconfluenthypergeometricLommelStruveunitdisk
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper supplies explicit parameter conditions under which three classical special functions—the normalized confluent hypergeometric function, the Lommel function of the first kind, and the generalized Struve function of the first kind—are exponential convex or exponential starlike in the unit disk. A function is exponential convex when $1+zf''(z)/f'(z)$ lies inside the image of the unit disk under $e^z$, and exponential starlike when $zf'(z)/f(z)$ does; equivalently, the logarithm of the corresponding quotient has modulus less than $1$. The conditions are simple inequalities in the parameters, for instance $(e-1)|c-2|+|a|\le (e-1)^2(e+1)/e^2$ for a normalized multiple of the confluent hypergeometric function, and each is proved by reducing a second-order differential equation to a subordination to $e^z$. These are sufficient conditions, so they enlarge the known explicit parameter ranges for geometric properties of these special-function families.

What carries the argument

The load-bearing mechanism is the admissibility criterion for subordination to the exponential function. A normalized analytic $f$ is exponential convex exactly when $p=1+zf''(z)/f'(z)$ is subordinate to $e^z$, and exponential starlike exactly when $p=zf'(z)/f(z)$ is; the paper writes both conditions as $p\in P_e$, where $P_e=\{p:p(0)=1,\ p(z)\prec e^z\}$. Lemma 1.1 states that if a differential expression $\Psi(r,s,t;z)$ avoids zero for every boundary candidate $r=e^{e^{i\theta}}$, $s=me^{i\theta}r$ with $m\ge 1$ and $\operatorname{Re}(1+t/s)\ge m(1+\cos\theta)$, then the analytic solution $p$ of $\Psi(p(z),zp'(z),z^2p''(z);z)=0$ must lie in $P_e$. The proofs convert the special function's differential equation into exactly such a $\Psi$ and then use elementary estimates—$|e^{e^{i\theta}}-1|\le e-1$, lower bounds on $\operatorname{Re}((e^{e^{i\theta}}-e^{-e^{i\theta}})e^{-i\theta})$ whose minimum is $2\sin 1$, and similar trigonometric extrema—to show the parameter inequalities force $|\Psi|>0$ on the boundary. Thus the inequalities are precisely the conditions under which the boundary of the exponential subordination class repels the differential expression.

What would settle it

Take a parameter pair that satisfies the hypotheses of Theorem 2.3 with equality in (2.3), such as $a=-0.5$ and $c=2-\frac{(e-1)^2(e+1)/e^2-0.5}{e-1}$, and compute $\sup_{|z|<1}|\log p(z)|$ for $p=1+z\Phi''(a;c;z)/\Phi'(a;c;z)$ on a fine grid in the unit disk; a single point with value above $1$ would disprove the membership claim for that parameter range. Alternatively, for the Lommel theorems, evaluate the cited lower bound $|h'_{\mu,\nu}(z)|\ge 1-4M/(N(2M-3))$ at a parameter pair meeting the first two hypotheses of Theorem 3.1 but lying near the boundary; finding a zero of $h'_{\mu,\nu}$ would break the analyticity step before subordination is applied.

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Extended reading notes

Core claim

The central claim is that, for each of the three families, a moderate order/size condition on the parameters together with one explicit inequality places the normalized function in the exponential classes. For the confluent hypergeometric function $\Phi(a;c;z)$, under $a>-1$, $c\ge a$ (or $a\le -1$, $c\ge \sqrt{1+(1+a)^2}$) and $(e-1)|c-2|+|a|\le (e-1)^2(e+1)/e^2$, the function $\frac{c}{a}(\Phi(a;c;z)-1)$ belongs to $K_e$; shifting the parameters by one turns this into $z\Phi(a;c;z)\in S_e^*$. For the normalized Lommel function $h_{\mu,\nu}$, condition (3.2) gives $h_{\mu,\nu}\in K_e$, condition (3.4) gives $f_{\mu,\nu}(z)=\int_0^z h_{\mu,\nu}(t)/t\,dt\in K_e$ and hence $h_{\mu,\nu}\in S_e^*$, and condition (3.7) gives $h_{\mu,\nu}(z)/z\in P_e$. For the normalized generalized Struve function $u_\nu$, condition (4.10) gives $u_\nu\in P_e$, while condition (4.13) gives $6\kappa(1-u_\nu)/c\in K_e$; the paper's final theorem shows this last membership is preserved under convolution with arbitrary convex functions, hence under the two standard integral transforms it applies. In each theorem the auxiliary function $p=1+zf''/f'$ or $p=zf'/f$ is shown to satisfy a second-order differential equation, and the parameter inequality is exactly what prevents the associated differential expression from vanishing on the boundary candidates of $P_e$.

Load-bearing premise

The argument depends on cited lower-bound estimates from the literature that guarantee the needed derivatives do not vanish and certain real parts stay positive under the stated parameter restrictions, for example estimates of the forms $|h'_{\mu,\nu}(z)|>0$, $\operatorname{Re}(h_{\mu,\nu}(z)/z)>0$, and $|u'_\nu(z)|>0$; if any of these estimates is invalid or its hypotheses are misapplied, the auxiliary function $p$ need not be analytic and the subordination argument cannot begin.

Editorial extensions

If this is right

  • If Theorem 2.3 is correct, every parameter pair satisfying (2.3) yields an explicit exponential convex function, and the shifted inequality in Corollary 2.4 yields an exponential starlike function, both with no further numerical work.
  • If Theorem 3.1 and Theorem 3.2 are correct, the normalized Lommel functions and their integral transforms supply two linked families in $K_e$ and $S_e^*$ for the parameter ranges in (3.2) and (3.4).
  • If Theorem 4.3 and Theorem 4.6 are correct, the generalized Struve functions give exponential convex functions that remain exponential convex after convolution with any convex function, including the Alexander and Libera integral operators.
  • Corollaries 4.4 and 4.5 translate the general Struve condition into a single explicit lower bound on the order $\nu$, so the normalized Struve and modified Struve functions are exponential convex and their derivatives exponential starlike above that bound.
  • Theorem 3.3 adds the derivative quotient $h_{\mu,\nu}(z)/z$ itself to the class $P_e$ under condition (3.7), which is a direct starlikeness-type statement for the Lommel family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same boundary-repulsion scheme should transfer to other normalized hypergeometric solutions of second-order differential equations; each new family only requires a second-order equation whose coefficients are polynomial in $z$ and a fresh set of trigonometric extrema.
  • Because the paper's inequalities are sufficient and not necessary, numerical exploration of the parameter space could identify larger regions; any widening would improve corollaries such as the explicit lower bound on $\nu$ in Corollary 4.4.
  • The convolution closure used in Theorem 4.6 suggests a cheap way to generate many exponential convex functions: convolve the special functions with arbitrary convex functions, so the parameter conditions imply whole families of examples rather than single functions.
  • Replacing the target function $e^z$ by another standard simply connected image domain would produce analogous classes; the estimates would change only through the boundary shape of the target domain.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript derives sufficient conditions on the parameters of the confluent hypergeometric function Phi(a;c;z), the Lommel function of the first kind h_{mu,nu}, and the generalized Struve function u_{nu} for these functions to lie in the exponential convex class K_e or the exponential starlike class S_e^*. The method is the differential-subordination admissibility lemma of Naz et al. [16]. Section 2 treats the confluent hypergeometric function, Section 3 the Lommel function, and Section 4 the generalized Struve function, with several examples and graphical illustrations.

Significance. If the results are correct, they provide new, explicit parameter ranges for exponential convexity and starlikeness of these special functions, and the derivations are analytic and parameter-free. The proofs in Sections 3 and 4 are internally consistent, and the constants in those admissibility estimates match the stated conditions. The paper does not rely on fitted parameters or on ad-hoc axioms tailored to these functions, since the key lemma is a published general criterion [16]. However, the central confluent-hypergeometric theorem is not in a usable form as printed: its statement is false as written, and its proof contains an algebraic inconsistency in the key estimate. Because this is the first and most prominent result, the manuscript requires substantial revision before it can be considered further.

major comments (3)
  1. [Section 2, Theorem 2.3] The theorem states that (Phi(a;c;z)-1)^{c/a} belongs to K_e, but the proof and Example 2.5 concern the function (c/a)(Phi-1). These are not the same: since Phi-1 has a simple zero at z=0, the power (Phi-1)^{c/a} is not single-valued and analytic near 0 when c/a is not an integer, and it is not normalized with f'(0)=1 when c/a is an integer different from 1. For instance, with a=1,c=2 the stated function is (Phi-1)^2, whose derivative at 0 is 0, so it cannot belong to K_e. The proof's p=1+zPhi''/Phi' is exactly the exponential-convexity expression for (c/a)(Phi-1), so the intended statement is evidently the constant-multiple version. The theorem must be restated accordingly.
  2. [Section 2, proof of Theorem 2.3 after (2.6)] The proof concludes |Psi| > 1/e - 1/e^2 + 1 - (e-1)|c-2| - |a| - e >= 0 whenever (2.3) holds. But 1/e - 1/e^2 + 1 - e = -(e - 1 - 1/e + 1/e^2) = -(e-1)^2(e+1)/e^2, so the right-hand side is always negative; subtracting the additional nonnegative terms makes it strictly negative. Thus the stated lower bound cannot be nonnegative under condition (2.3), and the claimed inequality has the wrong sign. Consequently the admissibility condition Psi notin Omega is not established. The assertion that the auxiliary function g attains its global minimum at theta=pi is also not justified by the second derivative test alone, since g depends on m>=1. The proof of Theorem 2.3 is therefore invalid as written.
  3. [Section 2, integral-representation paragraph] The paragraph defines g_delta(z):=Phi(1;1+delta;z) and h_delta(z):=zPhi(1;1+delta;z) and claims membership in K_e and S_e^*, respectively. But Phi(1;1+delta;0)=1, so Phi is not in the normalized class A, and zPhi is not in A either. What follows from the (corrected) Theorem 2.3 is, for a=1,c=1+delta, the membership of (1+delta)(Phi-1) in K_e, not of Phi itself. This passage should be rewritten to define the normalized functions and to state the correct conclusion.
minor comments (5)
  1. [Introduction] The phrase 'Milen conjecture' should be 'Milin conjecture'.
  2. [Section 3, Theorem 3.1] The proof uses 'using (3.2)' to assert that mu > 2, but this implication is not immediate from the displayed inequality and should be justified or stated as a separate consequence.
  3. [Sections 2 and 3] The symbol g(theta) is defined with different meanings in Theorem 2.3 and Theorem 3.1; this is confusing and the second function should be given a different name.
  4. [Example 2.5] Example 2.5 uses Lambda(1;2;z)=2(Phi(1;2;z)-1), which is inconsistent with the statement of Theorem 2.3 as printed but consistent with the intended corrected version; the text should be aligned after the theorem is fixed.
  5. [Throughout] The figures are described only by placeholder text; ensure the final version contains the actual graphics and that they are legible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proofs rely on a general admissible-function lemma and external lower-bound estimates; Theorem 2.3 has a statement/proof mismatch that is a correctness issue, not a circularity.

full rationale

The derivation chain is not circular. The central tool, Lemma 1.1, is quoted from the authors' earlier paper [16], but it is a general admissibility criterion for subordination to e^z: it is parameter-free, its hypotheses do not mention the confluent hypergeometric, Lommel, or Struve functions, and it is applied uniformly on the boundary set r=e^{e^{i\theta}}, s=me^{i\theta}r. The lower-bound inputs [12, Theorem 1, p. 336], [26, p. 1040], [26, Corollary 2.4, p. 1042], and [18, Equation 2.19, p. 10] are external published estimates with stated parameter restrictions; none is fitted in this paper and none presupposes the K_e or S_e^* conclusion. Each theorem reduces the desired subordination to a verified inequality |\Psi|>0 on that boundary set, and the conditions such as (2.3), (3.2), (3.4), (4.10), and (4.13) are sufficient inequalities, not fitted parameters or renamed predictions. The self-citations ([16] for Lemma 1.1 and [11] for convolution closure in Theorem 4.6) are published general results whose assumptions do not include the target special-function classes, so they do not make the argument circular. A separate correctness flag, not a circularity flag: Theorem 2.3 states (\Phi(a;c;z)-1)^{c/a}\in K_e, but the proof defines p(z)=1+z\Lambda''/\Lambda'=1+z\Phi''/\Phi', which is exactly the exponential-convexity expression for a constant multiple (c/a)(\Phi-1), not for the printed power; Example 2.5 likewise uses 2(\Phi(1;2;z)-1). For non-integer c/a the printed power is not analytic at 0. This internal mismatch and the sign issue in the estimate around (2.6) undermine the printed theorem as stated, but they are not a circular reduction of the conclusion to the hypotheses.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper adds no invented entities and fits no parameters. It leans on its own admissible-function lemma [16], positivity and nonvanishing bounds from Miller-Mocanu [12], Yagmur [26], and Orhan-Yagmur [18], and standard closure or duality results [11]. The most fragile input is the correctness of those cited bounds on the claimed parameter ranges.

assumptions (7)
  • standard math Lemma 1.1 (admissibility criterion for p ≺ e^z) from Naz, Nagpal, and Ravichandran [16].
    Used in every theorem as the engine to conclude p(z) ≺ e^z; not proved in this paper.
  • standard math Miller-Mocanu [12, Theorem 1, p.336] positivity of Re((c/a)Phi'(a;c;z)) under the stated conditions.
    Invoked in Theorem 2.3 to guarantee Phi' is nonzero and p is analytic; not proved here.
  • standard math Yagmur [26, p.1040] lower bound for |h'_mu_nu| and [26, Cor 2.4] positivity of Re(h_mu_nu/z).
    Used in Theorems 3.1 and 3.2 to guarantee nonvanishing derivatives; external geometric estimates.
  • standard math Orhan-Yagmur [18, Eq 2.19] lower bound for |u'_nu| and [18, Prop 2.1(v)] recursion for generalized Struve functions.
    Used in Theorems 4.1, 4.3, and Corollary 4.2.
  • standard math Closure of K_e and S_e^* under convolution with convex functions [11, Theorem 2.9].
    Used in Theorem 4.6 to transport convexity to Alexander and Libera transforms.
  • standard math Alexander duality: f in K_e if and only if zf' in S_e^*.
    Used to pass from exponential convexity to exponential starlikeness in Corollary 2.4.
  • standard math Kummer differential equation and derivative identity (a-1)Phi(a;c;z) = (c-1)Phi'(a-1;c-1;z); Lommel and Struve differential equations (3.1) and (4.9).
    Basis for the differential equations that p satisfies in each proof.

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Pith. "Pith review of Exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions." pith.science (2026). https://pith.science/paper/ZMPIDYVS

@misc{pith2026190807266,
  author       = {Pith},
  title        = {Pith review of: Exponential starlikeness and convexity of confluent hypergeometric, Lommel and Struve functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMPIDYVS}},
  note         = {Machine review of arXiv:1908.07266}
}
read the original abstract

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.

Figures

Figures reproduced from arXiv: 1908.07266 by the authors.

Figure 1
Figure 1. Graph showing qi(D) ⊂ exp(D). Similarly, the subordinations Φ(−25; 27; z) ≺ e z and Φ(−100; 102; z) ≺ e z are graphi￾cally shown in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Graph showing q(D) ⊂ exp(D) For Re(a) > 0 and Re(c) > 0, the function Φ(a; c; z) also has following the integral representation [13, Equation (1.2-8), p. 5]: Φ(a; c; z) = Γ(c) Γ(a) · Γ(c − a) Z 1 0 t a−1 (1 − t) c−a−1 e tzdt = Z 1 0 e tzdµ(t) where dµ(t) = Γ(c)t a−1 (1 − t) c−a−1 Γ(a) · Γ(c − a) dt is the probability measure on [0, 1]. Therefore using Theorem 2.3, we have gδ(z) := Φ(1; 1 + δ; z) = δ Z 1 0 (1 − t) δ−… view at source ↗

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