Coefficients of strongly alpha-convex and alpha-logarithmicaly convex functions

Authors

  • Derek Keith Thomas Swansea University

DOI:

https://doi.org/10.5556/j.tkjm.48.2017.2036

Keywords:

Univalent functions, inverse coefficients, strongly starlike and convex functions.

Abstract

Let the function $f$ be analytic in $D=\{z:|z|<1\}$ and be  given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$.  For $0< \beta \le 1$, denote by  $C (\beta)$ and $S^*(\beta)$ the classes of strongly  convex functions and strongly starlike functions respectively.  For $0\le \alpha \le1$ and $0< \beta \le 1$, let $M(\alpha, \beta)$ be the class of strongly alpha-convex functions defined by $\left|\arg \Big((1-\alpha) \dfrac{zf'(z)}{f(z)}\Big)+\alpha (1+\dfrac{zf''(z)}{f'(z)})^{}\Big)\right|< \dfrac{\pi \beta }{2}$, and  $M^{*}(\alpha, \beta)$ the class of strongly alpha-logarithmically  convex functions defined by  $\left|\arg\Big( \Big( \dfrac{zf'(z)}{f(z)}\Big)^{1-\alpha}\Big(1+\dfrac{zf''(z)}{f'(z)}\Big)^{\alpha}\Big)\right|< \dfrac{\pi \beta }{2}$.  We give sharp bounds for the initial coefficients of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$, and for the initial coefficients of the inverse function $f^{-1}$ of $f\in M(\alpha,\beta)$ and $f\in M^{*}(\alpha,\beta)$. These results generalise and unify known coefficient inequalities for $C (\beta)$ and $S^*(\beta)$

Author Biography

Derek Keith Thomas, Swansea University

Department ofMathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK.

References

R. M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malaysian Math. Soc., 26 (2003), 63--71.

R. M. Ali and V. A. Singh, On the fourth and fifth coefficients of strongly starlike functions, Results in Mathematics, 29(1996), 197--202.

D. A. Brannan, J. Clunie and W. E. Kirwan, Coefficient estimates for a class of starlike functions, Can. J. Math., XXII(1970), 476--485.

M. Darus and D. K. Thomas, $alpha$-logarithmically convex functions, Indian J. Pure. Appl. Math., 29(1998), 1049--1059.

M. Darus and D. K. Thomas, Inverse coefficients of $alpha$-logarithmically convex functions, Jnanabha, 45(2015), 31--36.

K. Kulshrestha, Coefficients for alpha-convex univalent functions, Bull. Amer. Math. Soc., 80(1974), 341--342.

Z. Lewandowski, S. S. Miller and E. J. Zlotkiewicz, Gamma-starlike functions, Ann. Univ. Marie-Curie Sklodowska, 27(1974), 53--58.

C. Lowner, Untersuchungen uber schlichte konforme Abbildungen des Einheitskreises, I, Math. Ann., 89(1923), 103--121.

W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceeding of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang and S. Zhang (Eds), Int. Press, (1990), 157--169.

S. S. Miller, P. Mocanu and M. 0. Read, All $alpha$-convex functions are univalent and starlike, Proc. Amer. Math. Soc.,37(1973), 553--554.

D. V. Prokhorov and J. Szynal, Inverse coefficients for $(alpha ,beta )$-convex functions, Annales Universitatis Mariae Curie - Sklodowska, X(1981), No.15, 125--141.

D. K. Thomas and S. Verma, Invariance of the coefficients of strongly convex functions, Bull. Australian Math, Soc., (2016),doi.10.1017/S0004972716000976..

P. Todorov, Explicit formulas for the coefficients of $alpha$ convex functions, $alpha ge0$, Can.J. Math., XXXIX (1987), 769--783.

Downloads

Published

2017-03-30

How to Cite

Thomas, D. K. (2017). Coefficients of strongly alpha-convex and alpha-logarithmicaly convex functions. Tamkang Journal of Mathematics, 48(1), 17–29. https://doi.org/10.5556/j.tkjm.48.2017.2036

Issue

Section

Papers