Convexity and starlikeness of functions satisfying an inequality of Mocanu type (Q2782469)

Let \(A\) be the class of analytic functions on the unit disk normalized by \(h(0)=h'(0)-1=0.\) In this paper the author gives upper bounds of \(\lambda_1\) and \(\lambda_2,\) so that for a function \(h\in A,\) the conditions \(\operatorname {Re}[z(z(zh'(z))'')']>-\lambda_1,\) \(|z|<1,\) and \(\operatorname {Re}[z(zh'(z))'']>-\lambda_2,\) \(|z|<1,\) imply starlikeness and convexity. Also, for a function satisfying the first condition, starlikeness of the Bernardi integral operator and upper bounds on \(|zf''(z)|\) and \(\operatorname {Re}[zf''(z)]\) are determined. The author makes use of a result on convolution (Hadamard product) from \textit{M. Rosihan}, \textit{S. Ponnusamy} and \textit{V. Singh} [Ann. Pol. Math. 61, No. 2, 135-140 (1995; Zbl 0818.30006)].NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].