Some geometric properties of a subclass of multivalent analytic functions defined by the first-order differential subordination

Jin-Lin Liu

Filomat, Tome 34 (2020) no. 9, p. 2819

Résumé

A new class Tn(A,B, λ) of multivalent analytic functions defined by the first-order differential subordination is introduced. Some geometric properties of this new class are investigated. The sharp lower bound on |z| = r 1 for the functional Re { (1 − λ) f (z)zp + λ f ′(z) pzp−1 } over the class Tn(A,B, 0) is given. 1. Introduction Throughout our present investigation, we assume that n, p ∈N, −1 ≤ B 1, B A and λ > 0. (1.1) LetAn(p) denote the class of functions of the form f (z) = zp + ∞∑ k=n akzk+p (1.2) which are analytic in the open unit diskU = {z : |z| 1}. For functions f (z) and 1(z) analytic in U, we say that f (z) is subordinate to 1(z) and write f (z) ≺ 1(z) (z ∈ U), if there exists an analytic function w(z) inU such that |w(z)| ≤ |z| and f (z) = 1(w(z)) (z ∈ U). If the function 1(z) is univalent inU, then f (z) ≺ 1(z) (z ∈ U)⇔ f (0) = 1(0) and f (U) ⊂ 1(U).

Détail

DOI : 10.2298/FIL2009819L

Classification : 30C45, 30C80
Keywords: Analytic function, First-order differential subordination, Coefficient estimate, Sharp bound