Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

Abstract

Let fk(z)=z+∑n=2kanzn{f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f(z)=z+∑n=2∞anznf\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}. In this paper, we determine sharp lower bounds for Re{f(z)/fk(z)}{\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\}, Re{fk(z)/f(z)}{\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\}, Re{f′(z)/fk′(z)}{\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re{fk′(z)/f′(z)}{\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\}, where f(z)f\left(z) belongs to the subclass Jp,qm(μ,α,β){{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean (p,q)\left(p,q)-differential operator. In addition, the inclusion relations involving Nδ(e){N}_{\delta }\left(e) of this generalized function class are considered.

Keywords

• analytic
• univalent
• (p, q)-differential operator
• partial sum
• inclusion relation
• 30c45
• 30c50