Preserving properties and pre-Schwarzian norms of nonlinear integral transforms
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PRESERVING PROPERTIES AND PRESCHWARZIAN NORMS OF NONLINEAR INTEGRAL TRANSFORMS S. KUMAR† and S. K. SAHOO∗ Discipline of Mathematics, Indian Institute of Technology Indore, Indore 453552, India e-mails: [email protected], [email protected] (Received September 8, 2019; accepted December 26, 2019)
Abstract. We study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of such integrals.
1. Introduction and preliminaries Let A denote the class of all analytic functions f in the unit disk D := {z ∈ C : |z| < 1} with the normalization f (0) = 0 and f ′ (0) = 1. The subclass of A consisting of all univalent functions is denoted by S. The notations S ∗ and K stand for the well-known classes of functions in S that are starlike (with respect to origin) and convex, respectively, see [3]. We also denote by C, the class of close-to-convex functions in D, i.e. functions f ∈ A satisfying ′ iα f (z) Re e ′ >0 g (z) for some g ∈ K and a real number α ∈ (−π/2, π/2) (see [4, Vol. 2, p. 2]). Some natural generalizations of the classes S ∗ and K are available in the literature. In this paper, we consider the following generalizations: zf ′ (z) (1.1) Sα∗ (λ) = f ∈ A : Re eiα > λ cos α , f (z) ∗ Corresponding
author. work of the first author is supported by CSIR, New Delhi (Grant No. 09/1022(0034)/2017-EMR-I). Key words and phrases: integral transform, Hornich operator, Ces` aro transform, preSchwarzian norm, univalent function, spirallike function, convex function, close-to-convex function. Mathematics Subject Classification: primary 30C55, 35A22, secondary 30C45, 35A23, 65R10. † The
c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, ´ Budapest 0236-5294/$20.00 Akade ´miai Kiado , Budapest, Hungary
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S. S. KUMAR KUMAR and and S. S. K. K. SAHOO SAHOO
and (1.2)
K(λ) =
zf ′′ (z) f ∈ A : Re +1 f ′ (z)
>λ ,
where α ∈ (−π/2, π/2) and λ < 1. Note that the class Sα∗ (λ) is known as the class of α-spirallike functions of order λ and the class K(λ) denotes the class of convex functions of order λ. Recall that the class Sα∗ (λ), for 0 ≤ λ < 1, is studied by several authors in different perspective (see, for instance, [4, p. 93, Vol. 2] and [13,22]). Further, the class K(λ), −1/2 ≤ λ < 1, is introduced, for instance, in [14] and references therein. Originally, a slight modification of this class was first studied by T. Umezawa [25] by characterizing with the class of functions convex in one direction. We can also easily observe that the class K(λ), −1/2 ≤ λ < 1, is contained in the class C that follows from Kaplan’s Theorem, see [3, §2.6]. Note that K(λ), 0 ≤ λ < 1, is the well-known class of normalized convex univalent functions. Recall from the literature that S ∗ (λ) := S0∗ (λ),
S ∗ := S ∗ (0) and K := K(0).
Motivation to consider the class S ∗ (λ), λ < 1, comes, for instance, from the classes S ∗ (−1/2) and K(−1/2) already studied i
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