Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
Copyright policy )Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {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 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( 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 J p , q m ( 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 N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.
- VIT-AP University India
- Chifeng University China (People's Republic of)
- University College of Engineering Tindivanam India
- Anna University, Chennai India
- Vellore Institute of Technology University India
Analytic Functions, 30c50, Complex Analysis, (p, q)-differential operator, Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), partial sum, QA1-939, FOS: Mathematics, Complex Analysis and Operator Theory, inclusion relation, Spectral Factorization, Applied Mathematics, Physics, Geometric Function Theory and Complex Analysis, Partial Differential Equations, 30c45, Quasiconformal Mappings, Nonlocal Partial Differential Equations and Boundary Value Problems, Combinatorics, Coefficient problems for univalent and multivalent functions of one complex variable, Physical Sciences, analytic, Geometry and Topology, \((p, q)\)-differential operator, univalent, Mathematics
Analytic Functions, 30c50, Complex Analysis, (p, q)-differential operator, Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), partial sum, QA1-939, FOS: Mathematics, Complex Analysis and Operator Theory, inclusion relation, Spectral Factorization, Applied Mathematics, Physics, Geometric Function Theory and Complex Analysis, Partial Differential Equations, 30c45, Quasiconformal Mappings, Nonlocal Partial Differential Equations and Boundary Value Problems, Combinatorics, Coefficient problems for univalent and multivalent functions of one complex variable, Physical Sciences, analytic, Geometry and Topology, \((p, q)\)-differential operator, univalent, Mathematics
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