Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc

Abstract

In the present paper we determine sharp lower bounds of the real part of the ratios of harmonic univalent meromorphic functions to their sequences of partial sums. Let H denote the class of functions f that are harmonic univalent and sense-preserving in U =; fz : jzj > 1g which are of the form f(z) = h(z) + g(z) ; where h(z) = z + 1X n=1 anz􀀀n ; g(z) = 1X n=1 bnz􀀀n. Now, we de ne the sequences of partial sums of functions f of the form fs(z) = z + Xs n=1 anz􀀀n + g(z); e fr(z) = g(z) + Xr n=1 bnz􀀀n; fs;r(z) = z + Xs n=1 anz􀀀n + Xr n=1 bnz􀀀n: In the present paper we will determine sharp lower bounds for Re n f(z) fs(z) o ; Re n fs(z) f(z) o ; Re n f(z) e fr(z) o ; Re n e fr(z) f(z) o ; Re n f(z) fs;r(z) o , Re n fs;r(z) f(z) o :