Certain subclass of analytic functions defined by means of differential subordination

Abstract

For ??(?,?], let Ra(?) denote the class of all normalized analytic functions in the open unit disk U satisfying the following differential subordination: f'(z)+1/2(1+ei?)z f''(z)<?(z) z ? U), where the function ?(z) is analytic in the open unit disk U such that ?(0)=1. In this paper, various integral and convolution characterizations, coefficient estimates and differential subordination results for functions belonging to the class R?(?) are investigated. The Fekete-Szeg? coefficient functional associated with the kth root transform [f(zk)]1/k of functions in R?(?) is obtained. A similar problem for a corresponding class R?,?(?) of bi-univalent functions is also considered. Connections with previous known results are pointed out.

The Principle of Differential Subordination and Its Application to Analytic and p-Valent Functions Defined by a Generalized Fractional Differintegral Operator

“…Hence, by applying Equations (25), (26) and (27) to Equation (24), we can find easily an upper bound for the right-hand side of Equation (24). Thus, the function I(z, t) satisfies the second condition of Lemma 5, which proves that I(z, t) is a subordination chain.…”

“…Our results give interesting new properties, and together with other papers that appeared in the last years could emphasize the perspective of the importance of differential subordinations and generalized fractional differintegral operators. We also note that, in recent years, several authors obtained many interesting results involving various linear and nonlinear operators associated with differential subordinations and their dual problrms (for details, see [21][22][23][24][25][26][27][28]).…”

Logarithmic Coefficients for Univalent Functions Defined by Subordination

“…Thus, every convex function is starlike but not conversely; however, each starlike function is convex in the disk of radius 2 − √ 3. Lately, several researchers have subsequently investigated similar problems in the direction of the logarithmic coefficients, the coefficient problems, and differential subordination [3][4][5][6][7][8][9][10][11], to mention a few. For example, the rotation of Koebe function k(z) = z(1 − e iθ ) −2 for each θ has logarithmic coefficients γ n = e iθn /n, n ≥ 1.…”

Majorization problems for certain classes of multivalent analytic functions related with the Srivastava-Khairnar-More operator and exponential function

“…Later, Roberston [17] (see also [19]) gave the concept of quasi-subordination as below. and say f is subordinate to in U, denoted by (see [21]; also see [18,22,23,29])…”