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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 \(A.\)

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.)