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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

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.

References

  1. B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan and S. M. Watt, Maple V language reference manual, Springer-Verlag, New York, 1991.
  2. A. Cordova and S. Ruschweyh, Subordination of polynomials, Rocky Mountain J. Math., 21(1)(1991), 159-170.
  3. E. Egervary, Abbildungseigenschaften der arithmetischen Mittel der geometrischen Reihe, Math. Z., 42(1)(1937), 221-230. https://doi.org/10.1007/BF01160074
  4. L. Fezer, On new properties of Arithmetical Means of the Partial Sums of Fourier Series, J. Math. Phys., 13(1-4)(1934), 1-17. https://doi.org/10.1002/sapm19341311
  5. Y. Komatu, On analytic prolongation of a family of operators, Mathematica (Cluj), no. 2, 32(55)(1990), 141-145.
  6. J. L. Lewis, Applications of a Convolution Theorem to Jacobi Polynomials, SIAM J. Math. Anal., 10(6)(1979), 1110-1120. https://doi.org/10.1137/0510102
  7. J. L. Lewis, Convexity of certain series, L. London Math. Soc. (2), 27(3)(1983), 435-446. https://doi.org/10.1112/jlms/s2-27.3.435
  8. G. Polya and I. J. Schoenberg, Remarks on de la Vallee Poussin means and convex conformal maps of the circle, Pacific J. Math., 8(2)(1958), 295-334. https://doi.org/10.2140/pjm.1958.8.295
  9. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Math. Soc. Monogr. 26, The Clarendon Press, Oxford University Press, Oxford, 2002, xiv+742 pp.
  10. M. S. Robertson, Power Series with Multiply Monotonic Coefficients, Michigan Math. J., 16(1969), 27-31.
  11. S. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv., 48(1973), 119-135. https://doi.org/10.1007/BF02566116
  12. S. Ruscheweyh, Geometric properties of the Ces'aro means, Results in Mathematics, 22(1992), 739-748. https://doi.org/10.1007/BF03323120
  13. S. Ruscheweyh, L. Salinas and T. Sugawa, Completely monotone sequences and universally prestarlike functions, Israel J. Math., 171(2009), 285-304. https://doi.org/10.1007/s11856-009-0050-9
  14. S. Ruscheweyh and L. C. Salinas, Subordination by Ces'aro means, Complex Variables Theory Appl., 21(1993), 279-285.
  15. S. Singh and R. Singh, Subordination by univalent functions, Proc. Amer. Math. Soc., 82(1)(1981), 39-47. https://doi.org/10.1090/S0002-9939-1981-0603598-1
  16. H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 12(1961), 689-693. https://doi.org/10.1090/S0002-9939-1961-0125214-5
  17. Wolfram Research, Inc., Mathematica, Version 11.0, Champaign, IL, 2016.
  18. M. Yadav, S. Gupta and S. Singh, Subordination of Cesaro means of convex functions, Bull. Malays. Math. Sci. Soc., 46(2)(2023), Paper No. 48, 17 pp.