Geometric properties of certain linear integral transforms

Abstract

For $\lambda>0$ and $0<\mu<n$, let ${\mathcal U}_n(\lambda,\mu)$ denote the class of all normalized analytic functions $f$ in the unit disc $\Delta$ of the form $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ such that $$\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right|<\lambda, ~ z\in \Delta, $$ where $n\in \mathbb N$ is fixed. In addition to the discussion of the basic properties of the class ${\mathcal U}_n(\lambda,\mu)$, we find conditions so that ${\mathcal U}_n(\lambda,\mu)$ is included in ${\mathcal S}_{\gamma}$, the class of all strongly starlike functions of order $\gamma$ $(0<\gamma\leq 1)$. We also find necessary conditions so that $f\in {\mathcal U}_n(\lambda,\mu)$ implies that $$\left|\frac{zf'(z)}{f(z)}-\frac{1}{2\beta}\right|<\frac{1}{2\beta},\quad \mbox{for all $z\in \Delta,$} $$ or $$\left|1+\frac{zf''(z)}{f'(z)}-\frac{1}{2\beta}\right| <\frac{1}{2\beta},\quad \mbox{for $|z|<r<1$},$$ where $r=r(\lambda,\mu,n)$ will be specified.