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

Lemma 1.1. [16] Let Ωbe a subset of C and the function Ψ: C3 × D →C satisfies the admissibility condition Ψ(r, s, t; z) /∈Ωwhenever r = eeiθ, s = meiθr and Re(1 + t/s) ≥m(1 + cos θ) where z ∈D, θ ∈[0, 2π) and m ≥1. If p is an analytic function in D with p(0) = 1 and Ψ(p(z), zp′(z), z2p′′(z); z) ∈Ωfor z ∈D, then p ∈Pe. It is worth to note that the admissiblity condition Ψ(r, s, t; z) ̸∈Ωis verified for all r = eeiθ, s = eeiθeeiθ and t with Re(1 + t/s) ≥0, that is, Re((t + s)e−iθe−eiθ) ≥0 for all θ

Theorem 2.1. Let the parameters a, c ∈C be constrained such that c is not a nonnegative integer and Re(c) ≥|a| + 2. Then Φ(a; c; z) ∈Pe.

Theorem 2.3. Let the parameters 0 ̸= a, c ∈R be constrained such that c is not a nonnegative integer and either (i) a > −1 and c ≥a or (ii) a ≤−1 and c ≥(1 + (1 + a)2)1/2. If such a and c satisfy the following condition: (e −1)|c −2| + |a| ≤(e −1)2(e + 1) e2 (2.3) then the function (Φ(a; c; z) −1)c/a ∈Ke.

Corollary 2.4. Let the parameters a, c ∈R be constrained such that c is not a nonnegative integer and either (i) a > 0 and c ≥a or (ii) a ≤0 and c ≥1 + (1 + a2)1/2. If such a and c satisfy the following condition: (e −1)|c −3| + |a −1| ≤(e −1)2(e + 1) e2 then the function zΦ(a; c; z) ∈S * e . Let us illustrate Theorem 2.3 and Corollary 2.4 by an example. Example 2.5. The constants a = 1 and c = 2 satisfy the conditions of Theorem 2.3. This gives Λ(1; 2; z) = 2(Φ(1; 2; z) −1) = 2 ez −z −1 z  ∈K

Corollary 2.4, the function hδ(z) := zΦ(1; 1 + δ; z) = δz Z 1 0 (1 −t)δ−1etzdt belongs to the class S * e when |δ −2| ≤(e2 −1)/e2.

Theorem 3.1. Let the parameters µ, ν ∈R be constrained such that µ ± ν are not negative odd integers, M = (µ + 5)2 −ν2 and N = (µ + 3)2 −ν2. If such µ and ν satisfy the following three conditions: µ > −5 + (3/2 + ν2)1/2, 4M N < 2M −3 and µ(1+2 sin(1))−1 4e(e−1) (µ + 1)(µ −7) −ν2 ≥e4−3e3+13e2 4 −3e 4 −3 e+2−2 sin(1) (3.2) then hµ,ν ∈Ke.

Lemma 1.1, we have p(z) ≺ez for all z ∈D which implies hµ,ν ∈Ke. □ Consider the Alexander transform fµ,ν : D →C of the function hµ,ν defined by fµ,ν(z) := Z z 0 hµ,ν(t) t dt.

Theorem 3.2. Let the parameters µ, ν ∈R be constrained such that µ±ν are not negative odd integers and (µ + 1)((µ + 1)(µ + 3) −ν2) ≥1/8. If such µ and ν satisfy the following condition: µ(2e −1) −1 4e(e −1) (µ −1)2 −ν2 ≥e3 −e2 + 13e 4 −4 (3.4) then the function fµ,ν ∈Ke and hence hµ,ν ∈S * e .

Lemma 1.1 to prove that p(z) ≺ez. For r = eeiθ, s = meiθeeiθ and Re((s+t)e−iθe−eiθ) ≥0

Theorem 3.3. Let the parameters µ, ν ∈C be constrained such that µ±ν are not negative odd integers. If such µ and ν satisfy the following condition: 4 Re(µ) ≥(e −1) (µ + 1)2 −ν2 −3 (3.7) then f′ µ,ν ∈Pe or hµ,ν(z)/z ∈Pe.

Lemma 1.1, note that |Ψ(r, s, t; z)| = |eeiθ| · (t + s)e−eiθ + (µ + 1)meiθ + 1 4(1 −e−eiθ)((µ + 1)2 −ν2) + z 4

Theorem 4.1. If the parameters κ, c ∈C are constrained such that κ is not a nonnegative integer and Re(κ) −1 2(e −1)|κ −1| ≥|c| 4 + 1 2 (4.10) then uν ∈Pe.

Corollary 4.2. If the parameters 0 ̸= c, κ ∈C are constrained such that κ is not a nonnegative integer and Re(κ + 1) −1 2(e −1)|κ| ≥|c| 4 + 1 2 then the function (2κ/cz)(1 −2zu′ ν(z) −uν(z)) ∈Pe. The next result deals with the sufficient condition on κ and c so that the generalized Struve function belongs to the class Ke.

Theorem 4.3. Let the parameters κ ∈R and 0 ̸= c ∈C be constrained such that κ is not a nonnegative integer and 2κ e (4 sin(1) + 3 −e) ≥(e + 1)|c| + 4(e −1)3 + 6(e −1)2 + 6 e −12 e2 (4.13) then 6κ(1 −uν)/c ∈Ke.

Corollary 4.4. Let the parameter ν ∈R be such that ν +3/2 is not a nonnegative integer and the function Hν : D →C be defined by Hν(z) = 2ν√πΓ  ν + 3 2  z−(ν+1)Hν(z) where Hν is the Struve function of the first kind of order ν defined in (4.2). If ν ≥ e 8 sin(1) + 6 −2e  4(e −1)3 + 6(e −1)2 + (e + 1) + 6 e −12

Corollary 4.5. Let the parameter ν ∈R and the function Lν : D →C be defined by (4.15). If the condition (4.14) holds, then the function 3(2ν + 3)(Lν −1) ∈Ke and 3(2ν + 3)zL′ ν ∈S * e . Let f and g be two analytic functions having the power series expansion as f(z) = z + P∞ n=1 an+1zn+1 and g(z) = z + P∞ n=1 bn+1zn+1 respectively. The Hadamard prod- uct f ∗g (or convolution) of f and g is defined as the power series (f ∗g)(z) = z + P∞ n=1 an+1bn+1zn+1. Note that both the classes Ke and S * e are cl

Theorem 4.6. If the parameters κ and c are constrained as in Theorem 4.3, then 6κ(1 −uν)/c ∗f ∈Ke for every f ∈K and therefore the functions A [6κ(1 −uν)/c] and L [6κ(1 −uν)/c] belong to the class Ke. References [1] R. M. Ali, S. R. Mondal and V. Ravichandran, On the Janowski convexity and starlikeness of the confluent hypergeometric function, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 2, 227–250. [2] A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73