SUFFICIENT CONDITIONS FOR STARLIKENESS

Title & Authors
SUFFICIENT CONDITIONS FOR STARLIKENESS
RAVICHANDRAN, V.; SHARMA, KANIKA;

Abstract
We obtain the conditions on $\small{{\beta}}$ so that $\small{1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3}$ implies p(z) $\small{{\prec}}$ (2+z)/(2-z), $\small{1+(1-{\alpha})z}$, $\small{(1+(1-2{\alpha})z)/(1-z)}$, ($\small{0{\leq}{\alpha}}$<1), exp(z) or $\small{{\sqrt{1+z}}}$. Similar results are obtained by considering the expressions $\small{1+{\beta}zp^{\prime}(z)/p(z)}$, $\small{1+{\beta}zp^{\prime}(z)/p^2(z)}$ and $\small{p(z)+{\beta}zp^{\prime}(z)/p(z)}$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition $\small{{\mid}log(zf^{\prime}(z)/f(z)){\mid}}$ < 1 or $\small{{\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $\small{(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0}$.
Keywords
convex and starlike functions;lemniscate of Bernoulli;subordination;cardioid;
Language
English
Cited by
1.
Subordinations for Functions with Positive Real Part, Complex Analysis and Operator Theory, 2017
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