# zbMATH — the first resource for mathematics

Differential sandwich theorems for certain analytic functions. (English) Zbl 1074.30022
Let $$H$$ be the class of analytic functions in the unit disk $$\Delta$$ and let $$H(a,n)$$ be the subclass of $$H$$ consisting of functions of the form $$f(z)=a+a_{n}z^{n}+a_{n+1}z^{n+1}+\dots$$. Let $$A$$ be the class of all analytic functions $$f(z)=z+a_{2}z^{2}+\dots$$. Let $$p$$, $$h\in H$$ and let $$\phi(r,s,t;z):C^3\times\Delta\to C$$. If $$p$$ and $$\phi(p(z),zp'(z),z^2p''(z);z)$$ are univalent and if $$p$$ satisfies the second order superordination $h(z)\prec\phi(p(z),zp'(z),z^2p''(z);z),$ then $$p$$ is a solution of the differential superordination above. The authors gave some applications of first order differential superordination for a function in $$A.$$

##### MSC:
 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)