Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Polylogarithms and Subordination of Some Cubic Polynomials

  • Manju Yadav (Department of Mathematics, Sant Longowal Institute of Engineering and Technology) ;
  • Sushma Gupta (Department of Mathematics, Sant Longowal Institute of Engineering and Technology) ;
  • Sukhjit Singh (Department of Mathematics, Sant Longowal Institute of Engineering and Technology)
  • Received : 2023.01.20
  • Accepted : 2023.09.15
  • Published : 2024.03.31
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Abstract

Let V3(z, f) and 𝜎(1)3(z, f) be the cubic polynomials representing, respectively, the 3rd de la Vallée Poussin mean and the 3rd Cesàro mean of order 1 of a power series f(z). If 𝒦 denotes the usual class of convex univalent functions in the open unit disk centered at the origin, we show that, in general, V3(z, f) ⊀ 𝜎(1)3(z,f), for all f ∈ 𝒦. Making use of polylogarithms, we identify a transformation, Λ : 𝒦 → 𝒦, such that V3(z, Λ(f)) ≺ 𝜎(1)3(z, Λ(f)) for all f ∈ 𝒦. Here '≺' stands for subordination between two analytic functions.

Keywords

Acknowledgement

The first author, Manju Yadav, acknowledges financial support from CSIR-UGC, Govt. of India, in the form of JRF vide Award Letter No. 211610111581.

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