On an inequality of the subclass of a class of analytic functions related prestarlike functions of \(\beta\) (Q2735424)

Let \(A\) be the class of functions \(f(z)=z+ a_2z^2+ \cdots\) analytic in the unit disk \(|z|<1\), and let \(S^*(\beta)\) be the class of usual starlike functions of order \(\beta\), \(0<\beta<1\). For \(\lambda>-1\) let \(D^\lambda f(z)= {z\over(1-z)^{\lambda+1}} *f(z)\), where \(*\) denotes convolution. Let \(J(\lambda, \alpha, \beta)\), \(\lambda>-1\), \(0\leq \alpha<1\), \(0\leq\beta \leq 1\), be the class of functions \(f\in A\) satisfying the conditions \({f(z)\over z} f'(z)\neq 0\) and NEWLINE\[NEWLINE\text{Re} \left[(1-\alpha) {z(D^\lambda f)'(z) \over D^\lambda f(z)}+ \alpha\left(1+ {z(D^\lambda f)'' (z)\over(D^\lambda f)'(z)} \right) \right] >\beta,\;|z|<1.NEWLINE\]NEWLINE Clearly, \(J(0,\alpha, \beta)=J(\alpha, \beta)\) is the class of \(\alpha\)-convex functions of order \(\beta\) and \(J(1-2 \beta,0,\beta)\) is the class of prestarlike functions of order \(\beta\). The main result of this paper is as follows: If \(f\in J(\lambda,\alpha,\beta)\), \(0< \alpha \leq\beta <2\alpha<1\), \(0<2\mu (1-\beta)\leq \alpha\), then NEWLINE\[NEWLINE\left [\left( {D^\lambda f(z) \over z}\right)^{{1 \over\alpha}-1} (D^\lambda f)'(z)\right]^\mu \prec (1-z)^{-{2 \mu\over \alpha}(1- \beta)},\;|z|<1,NEWLINE\]NEWLINE where \(\prec\) denotes subordination. As a consequence, the author obtains the inequalities NEWLINE\[NEWLINE\text{Re}\left[ \left( {D^\lambda f(z) \over z}\right)^{{1 \over\alpha}-1} (D^\lambda f)'(z) \right]^\mu >2^{-{2 \mu\over \alpha} (1-\beta)},\;|z|<1,NEWLINE\]NEWLINE for \(f\in J(\lambda,\alpha, \beta)\) and NEWLINE\[NEWLINE\text{Re} \left[\left( {f(z) \over z}\right)^{{1 \over\alpha}- 1}f'(z)\right]^\mu >2^{-{2 \mu\over \alpha}(1-\beta)},\;|z|<1NEWLINE\]NEWLINE for \(f\in J(\alpha,\beta)\).