Some applications of differential subordination for certain starlike functions

Authors: RAHIM KARGAR, LUCYNA TROJNAR SPELINA

Abstract: Let $\mathcal{S}^*(q_c)$ denote the class of functions $f$ analytic in the open unit disc $\Delta$, normalized by the condition $f(0)=0=f'(0)-1$ and satisfying the following inequality $\left|\left(\frac{zf'(z)}{f(z)}\right)^2-1\right| < c \quad (z\in\Delta, 0<c\leq1).$ By use of the subordination principle for the univalent functions we have $f\in\mathcal{S}^*(q_c)\Leftrightarrow \frac{zf'(z)}{f(z)}\prec \sqrt{1+cz} \quad(z\in\Delta, 0 <c\leq1).$ In the present paper, for an analytic function $p$ in $\Delta$ with $p(0)=1$ we give some conditions which imply $p(z)\prec \sqrt{1+cz}$. These conditions are then used to obtain some corollaries for certain subclasses of analytic functions.

Keywords: Analytic, univalent, subordination, Janowski starlike functions, Bernoulli lemniscate

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