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

Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator AbstractLet 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

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

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Publisher
de Gruyter
Copyright
© 2021 Huo Tang et al., published by De Gruyter
ISSN
2391-5455
eISSN
2391-5455
DOI
10.1515/math-2021-0028
Publisher site
See Article on Publisher Site

Abstract

AbstractLet 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.

Journal

Open Mathematicsde Gruyter

Published: May 20, 2021

Keywords: analytic; univalent; ( p , q )-differential operator; partial sum; inclusion relation; 30C45; 30C50

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