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Abstract
In this work,we provide upper estimate for Fekete-Szego functional and Second Hankel determinant for
the class 𝑆∗{𝑞} consisting of functions analytic in the unit disk U = {z∈ C : |z| < 1} and normalized by
f (0) =𝑓′(0) = 1 = 0 and which satisfies the subordination condition
Definitions (1)
Def 1.1
Definition 1.1. An analytic function f(z) is said to be subordinate to the analytic function g(z) and we write if there exist a function…
Definition 1.1. An analytic function f(z) is said to be subordinate to the analytic function g(z) and we write $f \prec g$ if there exist a function $\omega(z)$ analytic in U such that $\omega(0) = 1$ , $|\omega(z)| < 1$ and $f(z) = g(\omega(z))$ . If g(z) is univalent in U then $f \prec g \iff f(0) = g(0)$ and $f(|z| < 1) \subset g(|z| < 1)$
A function f(z) in S is said to be starlike in U if f(z) maps U onto a starlike domain with respect to $\omega_0 = 0$ . An analytic function f(z) is starlike in U if and only if $Re\frac{zf'(z)}{f(z)} > 0$ , $z \in U$ .
The class of starlike functions in U is denoted as $S$ . We define the class $S^\{q\}$ to be the class of functions $f \in A$ satisfying
$$\frac{zf'(z)}{f(z)} \prec \sqrt{1+z^2} + z = q(z) \qquad z \in U$$
where q(0)=1,
Furthermore let $\Omega$ be the class of analytic functions of the form $\omega(z) = \omega_1 z + \omega_2 z^2 + \omega_3 z^3 + \ldots$ in the unit disk U satisfying the condition $|\omega(z)| < 1$ .