Starlike and convexity properties for $p$-valent solutions of the Shah differential equation

Abstract

The starlikeness and the convexity in the unit disc and the growth of an entire function f (z) = z p + ∑ ∞ n=p+1 f n z n , p ∈ N, satisfying the differential equation z 2 w ′′ + (β 0 z 2 + β 1 z)w ′ + +(γ 0 z 2 + γ 1 z + γ 2)w = 0 (β 0 , β 1 , γ 0 , γ 1 , γ 2 are complex parameters) are studied. 1. Introduction. An analytic function univalent in D = {z : |z| < 1} f (z) = ∞ ∑ k=0 f k z k (1) is said to be convex if f (D) is a convex domain. It is well known [1, p. 203] that the condition Re {1+zf ′′ (z)/f ′ (z)} > 0 (z ∈ D) is necessary and sufficient for the convexity of f. According to W. Kaplan ([2]), a function f is said to be close-to-convex in D (see also [1, p. 583]) if there exists a convex in D function Φ such that Re (f ′ (z)/Φ ′ (z)) > 0 (z ∈ D). Every close-to-convex function in D is univalent in D and, therefore, f ′ (0) ̸ = 0. Hence it follows that the function f is close-to-convex in D if and only if the function (f (z) − f (0))/f ′ (0) is close-to-convex in D. It is clear that (f (z) − f (0))/f ′ (0) = g(z), where g(z) = z + ∞ ∑ k=2 g k z k (2) and g k = f k /f 1. The function g is said to be starlike if Re {zg ′ (z)/g(z)} > 0 (z ∈ D). S.M. Shah [3] indicated conditions on real parameters β 0 , β 1 , γ 0 , γ 1 , γ 2 of the differential equation z 2 w ′′ + (β 0 z 2 + β 1 z)w ′ + (γ 0 z 2 + γ 1 z + γ 2)w = 0, (3) under which there exists an entire transcendental solution given by (1) such that f and all its derivatives are close-to-convex in D. In particular he obtained the following result: if β 1 +γ 2 = 0, −1 ≤ β 0 < 0, β 1 > 0, γ 0 = 0 and −β 1 /2 < γ 1 ≤ 0, then equation (3) has an entire solution (2) such that all g (n) (n ≥ 0) are close-to-convex in D and ln M g (r) = (1 + o(1))|β 0 |r as r → +∞, where M g (r) = max{|g(z)| : |z| = r}.

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