Coefficients of strongly alpha-convex and alpha-logarithmicaly convex functions

Abstract

Let the function $f$ be analytic in $D=\{z:|z|<1\}$ and be  given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$.  For $0< \beta \le 1$, denote by  $C (\beta)$ and $S^*(\beta)$ the classes of strongly  convex functions and strongly starlike functions respectively.  For $0\le \alpha \le1$ and $0< \beta \le 1$, let $M(\alpha, \beta)$ be the class of strongly alpha-convex functions defined by $\left|\arg \Big((1-\alpha) \dfrac{zf'(z)}{f(z)}\Big)+\alpha (1+\dfrac{zf''(z)}{f'(z)})^{}\Big)\right|< \dfrac{\pi \beta }{2}$, and  $M^{*}(\alpha, \beta)$ the class of strongly alpha-logarithmically  convex functions defined by  $\left|\arg\Big( \Big( \dfrac{zf'(z)}{f(z)}\Big)^{1-\alpha}\Big(1+\dfrac{zf''(z)}{f'(z)}\Big)^{\alpha}\Big)\right|< \dfrac{\pi \beta }{2}$.  We give sharp bounds for the initial coefficients of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$, and for the initial coefficients of the inverse function $f^{-1}$ of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$. These results generalise and unify known coefficient inequalities for $C (\beta)$ and $S^*(\beta)$