A class of univalent functions defined by a differential inequality

Abstract

Let $\mathscr{A}$ be the class of analytic functions in the unit disk D with the normalization f(0) = f′(0) − 1 = 0. For λ > 0, denote by $\mathscr{M}$(λ) the class of functions f $\in$ $\mathscr{A}$ which satisfy the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq \lambda,\quad z\in \mathbf{D}.$$ We show that functions in $\mathscr{M}$(1) are univalent in D and we present one parameter family of functions in $\mathscr{M}$(1) that are also starlike in D. In addition to certain inclusion results, we also present characterization formula, necessary and sufficient coefficient conditions for functions in $\mathscr{M}$(λ), and a radius property of $\mathscr{M}$(1).