Maximal area integral problem for a family of multivalent functions of starlike type

Articles in Press

Document Type : Research Paper

Authors
C Infantina Sharmi
S Sunil Varma
Department of Mathematics, Madras Christian College, Tambaram, Chennai-600059, Tamil Nadu, India
10.22075/ijnaa.2025.34134.5095
Abstract
For a normalized analytic multivalent function $f$ defined in the open unit disk $\{z \in \mathbb{C}:|z|<1\},$ let $\Delta(r,f)$ and $L(r,f,p)$ denote the Dirichlet integral and the integral mean of $f$, respectively. In this research article, we consider a subclass $S^{*}_{p}(A,B,\lambda)$ of normalized analytic multivalent functions in the open unit disk and solve the Yamashitha's conjecture for this subclass. Also, we determine the extremal function for which the integral mean $L(r,f,p)$ is maximum.
Keywords
multivalent analytic function
subordination
integral means
area integral
Gaussian hypergeometric functions
[1] G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, U.K., 1999.
[2] M.K. Aouf, Coefficient estimates for some classes of p‑valent functions, Int. J. Math. Math. Sci. 11 (1988), no. 1, 47-54.
[3] G.R. Blakley, Classes of p‑valent starlike functions, Proc. Amer. Math. Soc. 13 (1962), 152-157.
[4] P.L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften, 259, New York, Berlin, Heidelberg, Tokyo, Springer‑Verlag, 1983.
[5] J. Clunie, On meromorphic schlicht functions, J. Lond. Math. Soc. 34 (1959), 215-216.
[6] L. de Branges, A proof of the Bieberbach's conjecture, Acta. Math. 154 (1985), 137-152.
[7] A.W. Goodman, On the Schwarz‑Christoffel transformation and p‑valent functions, Trans. Amer. Math. Soc. 68 (1950), 204-223.
[8] D.J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. 52 (1975), 191-195.
[9] W. Janowski, Some extremal problems for certain families of Analytic Functions, Ann. Polo. Math. 28 (1973), 297-326.
[10] M. Obradovic, S. Ponnusamy, and K‑J. Wirths, A Proof Of Yamashitha's Conjecture On Area Integral, Comp. Meth. Funct. Theory 13 (2013), 479-492.
[11] M. Obradovic, S. Ponnusamy, and K‑J. Wirths, Integral means and Dirichlet integral for analytic functions, Math. Nachr. 288 (2015), no. 2-3, 334-342.
[12] S. Ponnusamy and K.J. Wirths, On the problem Of Gromova and Vasilev on integral means and Yamashita's conjecture for spirallike functions, Ann. Acad. Sci. Fenn. 39 (2014), 721-731.
[13] S. Ponnusamy, S.K. Sahoo, and N.L. Sharma, Maximal area integral problem for certain class of univalent analytic functions, Medit. J. Math. 13 (2016), no. 2, 607-623.
[14] E.D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
[15] W. Rogosinski, On the coefficients of subordinate functions, Proc. Lond. Math. Soc. 48 (1943), no. 2, 48-82.
[16] S.K. Sahoo and N.L. Sharma, On maximal area integral problem for analytic functions in the starlike family, J. Classical. Anal. 6 (2015), no. 1, 73-84.
[17] N.L. Sharma, A note on coefficient estimates for some classes of p‑valent functions, Ukrain. Math. J. 70 (2018), no. 4, 549-563.
[18] N.L. Sharma, Integral means and maximum area integral problems for certain family of p‑valent functions, J. Korean. Math. Soc. 54 (2017), 1641-1665.
[19] N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley‑Interscience, New York, 1996.
[20] S. Yamashita, Area and length maxima for univalent functions, Bull. Aust. Math. Soc. 41 (1990), 435-439.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 01 August 2026

  • Receive Date 14 May 2024
  • Revise Date 13 April 2025
  • Accept Date 16 April 2025