Certain new subclass of close-to-convex harmonic functions defined by a third-order differential inequality

Abstract

In this paper, we define a new class R_H^(0,λ,δ) (L,M)   of normalized harmonic functions in the open unit disk  U={z∈C:|z|<1} which satisfying the following third-order differential inequality

Re(s^' (z)+λzs^'' (z)+δz^2 s^''' (z)-((L-1)/2M))>|u^' (z)+λzu^'' (z)+δz^2 u^''' (z)|,

Where,  λ≥δ≥0,1≤L<M≤-1,M≠0. First of all, we prove one-to-one correspondence between class R_H^(λ,δ) (L,M)   of analytic functions and class R_H^(0,λ,δ) (L,M)  of harmonic functions. Next we prove that every functions belonging to this class is closed-to-convex in open unit disk U. Furthermore, we investigate various properties of the class R_H^(0,λ,δ) (L,M) , such as coefficient bounds, growth estimates, sufficient coefficient condition. We establish that this class is closed under convex combination and convolution. We involve Gaussian hypergeometric function to discuss some applications of newly defined class of harmonic functions and construct harmonic polynomials which belong to the considered class R_H^(0,λ,δ) (L,M).   We also explore some new and known lemmas to prove our main results.