SUFFICIENT CONDITIONS FOR STARLIKENESS

Abstract

We obtain the conditions on <TEX>${\beta}$</TEX> so that <TEX>$1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$</TEX> implies p(z) <TEX>${\prec}$</TEX> (2+z)/(2-z), <TEX>$1+(1-{\alpha})z$</TEX>, <TEX>$(1+(1-2{\alpha})z)/(1-z)$</TEX>, (<TEX>$0{\leq}{\alpha}$</TEX><1), exp(z) or <TEX>${\sqrt{1+z}}$</TEX>. Similar results are obtained by considering the expressions <TEX>$1+{\beta}zp^{\prime}(z)/p(z)$</TEX>, <TEX>$1+{\beta}zp^{\prime}(z)/p^2(z)$</TEX> and <TEX>$p(z)+{\beta}zp^{\prime}(z)/p(z)$</TEX>. 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 <TEX>${\mid}log(zf^{\prime}(z)/f(z)){\mid}$</TEX> < 1 or <TEX>${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$</TEX> < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid <TEX>$(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$</TEX>.