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Solution of logarithmic coefficients conjectures for some classes of convex functions

  • Ebrahim Analouei Adegani EMAIL logo , Teodor Bulboacă , Nafya Hameed Mohammed and Paweł Zaprawa
Published/Copyright: February 15, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 1

Abstract

In [Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions, J. Funct. Spaces 2021 (2021), Art. ID 6690027], Alimohammadi et al. presented a few conjectures for the logarithmic coefficients γn of the functions f belonging to some well-known classes like C(1+αz) for α ∈ (0, 1], and CVhpl(1/2) . For example, it is conjectured that if the function f∈C(1+αz) , then the logarithmic coefficients of f satisfy the inequalities

|γn|≤α2n(n+1),n∈N.

Equality is attained for the function Lα, n, that is,

logLα,n(z)z=2∑n=1∞γn(Lα,n)zn=αn(n+1)zn+…,z∈U.

The aim of this paper is to confirm that these conjectures hold for the coefficient γn0−1 whenever the function f has the form f(z)=z+∑k=n0∞akzk , z∈U for some n0∈N , n0⩾2.

Keywords: Univalent functions; starlike and convex functions of some order; Faber polynomial; subordination; logarithmic coefficients
MSC 2010: Primary 30C45; Secondary 30C50; 30C80

  1. (Communicated by Marek Balcerzak )

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Received: 2021-10-18
Accepted: 2022-02-03
Published Online: 2023-02-15
Published in Print: 2023-02-23

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0009
Keywords for this article
Univalent functions; starlike and convex functions of some order; Faber polynomial; subordination; logarithmic coefficients

Articles in the same Issue

  1. RNDr. Stanislav Jakubec, DrSc. passed away
  2. Schur m-power convexity for general geometric Bonferroni mean of multiple parameters and comparison inequalities between several means
  3. Conditions forcing the existence of relative complements in lattices and posets
  4. Radically principal MV-algebras
  5. A topological duality for dcpos
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  7. Addendum to “A generalization of a result on the sum of element orders of a finite group”
  8. Remarks on w-distances and metric-preserving functions
  9. Solution of logarithmic coefficients conjectures for some classes of convex functions
  10. Multiple periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian and partially periodic potentials
  11. Asymptotic stability of nonlinear neutral delay integro-differential equations
  12. On Catalan ideal convergent sequence spaces via fuzzy norm
  13. Fourier transform inversion: Bounded variation, polynomial growth, Henstock–Stieltjes integration
  14. (ε, A)-approximate numerical radius orthogonality and numerical radius derivative
  15. Wg-Drazin-star operator and its dual
  16. On the Lupaş q-transform of unbounded functions
  17. K-contact and (k, μ)-contact metric as a generalized η-Ricci soliton
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