Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
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}\tex...
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| Autores principales: | , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
De Gruyter
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/5e4a1d6bcd1a43eb8bd7d9e85a3795b1 |
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| Sumario: | 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. |
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