Differential subordinations obtained by using a generalization of Marx-Strohhäcker theorem (Q2794907)

Let \(U\) be the complex unit disc and \(A_n\) the class of functions \(f:U\to \mathbb C\) having the form \(f(z)=z+\sum_{k=n+1}^\infty a_kz^k\). Using a differential subordination technique of S. S. Miller and P. T. Mocanu, the authors find some results like the following.NEWLINENEWLINENEWLINEFor all \(n\geq 3, \gamma>1, f\in A_n\) with NEWLINE\[CARRIAGE_RETURNNEWLINE\mathrm{Re}\left(\frac{zf''(z)}{f'(z)}+1\right)>-\frac{1}{2\gamma}\,\;\text{ in }\;\, U,CARRIAGE_RETURNNEWLINE\]NEWLINE the function \(f\) satisfies also NEWLINE\[CARRIAGE_RETURNNEWLINE\mathrm{Re}\frac{zf'(z)}{f(z)}>\frac{1}{2}\;\;\text{and}\;\;\mathrm{Re}\frac{f(z)}{z}>\frac{1}{2}\leqno (1)CARRIAGE_RETURNNEWLINE\]NEWLINE and NEWLINE\[CARRIAGE_RETURNNEWLINE\mathrm{Re}\sqrt{f'(z)}>\frac{1}{2}.\leqno(2)CARRIAGE_RETURNNEWLINE\]NEWLINE These are generalizations of the Marx and Strohhäcker theorem ((1)) and of the Komatu theorem ((2)), where the condition on \(f\) is similar to the above one, but with \(0\) instead of \(-1/2\gamma\).