Fekete-Szegö Estimate for a Class of Starlike Functions Involving Certain Analytic Multiplier Transform

Deborah Olufunmialyo Makinde¹, A. S. Oyekunle¹, T. O. Opoola²

¹Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria.
²Department of Mathematics, University of Ilorin, Ilorin, Nigeria.

DOI: 10.4236/jamp.2020.83041                

Abstract

In this paper, we investigated the coefficient estimates and the Fekete-Szegö problem for the subclass of analytic univalent functions involving the linear transformation D^s_α,β,γf for the normalized analytic function f (z) = z + a₂z² + a₃z³ + … .

Keywords

Analytic, Univalent, Starlike, Linear Transformation, Coefficient Estimates, Fekete-Szegö Inequality

1. Introduction

For the normalized analytic function f of the form:

$f\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots \mathrm{,}{a}_n\in C$ (1)

in the unit disk $U=\left\{z:\left|z\right|<1\right\}$, Fekete and Szegö [1], proved that,

$\left|a_3-\lambda a_2^2\right|\le 1+2{\text{e}}^{-\frac{2\lambda }{1-\lambda }},0<\lambda \le 1.$ (2)

And for the Schwarzian derivative $S_f$ given by

$S_f=\left(\frac{f^{″}}{f}\right)-\frac{1}{2}{\left(\frac{f^{″}}{f^{\prime }}\right)}^2=\frac{f^{‴}}{f^{\prime }}-\frac{3}{2}{\left(\frac{f^{″}}{f^{\prime }}\right)}^2$

Simple calculation shows that the coefficient functional ${\varphi }_f\left(\lambda \right)=a_3-\lambda a_2^2$ is related to the Schwarzian derivative by

${\varphi }_f\left(\lambda \right)=a_3-\lambda a_2^2=\frac{1}{6}\left(f^{‴}\left(0\right)-\frac{3\lambda }{2}{\left(f^{″}\left(0\right)\right)}^2\right)$

on normalized analytic functions f in the unit disk. Kanas and Darwish [2]

remarked that, when $\lambda =1,{\varphi }_f\left(\lambda \right)=a_3-a_2^2$, becomes $\frac{S_f\left(0\right)}{6}$, where $S_f$ denotes the Schwarzian derivative given in Equation (3) and that if we consider the nth root transformation

${\left(f\left(z^n\right)\right)}^{\frac{1}{n}}=z+c_{n+1}{z}^{2n+1}{c}_{n+1}{z}^{2n+1}+\cdots$

of the function in Equation (1), then $c_{n+1}=\frac{a_2}{2}$ and $c_{2n+1}=\frac{a_3}{n}+\frac{\left(n-1\right)a_2^2}{2n^2}$, so that

$a_3-\lambda a_2^2=n\left(c_{2n+1}-\mu c_{n+1}^2\right)$

where $\mu =\lambda n+\left(n-1\right)/2$. Several authors have discussed the nature of ${\varphi }_f\left(\lambda \right)$ for the normalized univalent functions in the unit disk. This is known as Fekkete-Szegö problem. Several authors have discussed the nature of $\varphi \left(f\right)$ for classes of normalized univalent functions in the unit disk and this is known as Fekkete-Szegö problem. For example, Choi, Kim and Sugawa [3], gave a generalized prestarlike function, while in [4] Fekete-Szegö problem was solved using subordination principle. Moreover, in [5] [6] [7] [8] and [9] Fekete-Szegö problems were solved for class of close-to-convex functions. Authors in [10] [11] [12] and [13] also solved Fekete-Szegö for classes of normalized analytic functions.

Now, we denote by S, the set of all functions of the form (1) that are normalized analytic and univalent in the unit disk $U=\left\{z:\left|z\right|<1\right\}$. Let $S^{\mathrm{<mo>\; *\; </mo>}}\left(\alpha \right)\mathrm{,}{S}^c\left(\alpha \right)$ be the classes of starlike and convex univalent function of order $\alpha$, of the form:

Now, we denote by S, the set of all functions of the form (1) that are normalized analytic and univalent in the unit disk $U=\left\{z:\left|z\right|<1\right\}$. Let $S^{\mathrm{<mo>\; *\; </mo>}}\left(\alpha \right)\mathrm{,}{S}^c\left(\alpha \right)$ be the classes of starlike and convex univalent function of order $\alpha$, of the form:

$S^{*}=\left\{f\in S:\mathrm{Re}\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\beta ,0\le \beta <1,z\in U\right\}$ (4)

Several authors have generalized notions of $\alpha$ -starlikeness and $\alpha$ -convexity onto a complex order $\alpha$ see [14] [15] [16]. When $\alpha =0$ in Equations (4) and (5), the starlike, respectively, convex functions with respect to the origin are obtained. With the aid of Ruscheweyh derivative, Kumar et al. [17] introduced the class $S_n\left(b\right)$ of functions $f\in S$ as follows:

Definition 1 Let b be a nonzero complex number, and let f be a univalent function of the form (1) such that $D^{n}f\left(z\right)\ne 0$ for $z\in U-\left\{0\right\}$. We say that f belongs to $S_n\left(b\right)$ if

$\mathrm{Re}\left\{1+\frac{1}{b}\left(\frac{z{\left(D^{n}f\right)}^{\prime }\left(z\right)}{D^{n}f\left(z\right)}-1\right)\right\}>0$ (5)

Moreover, the author in [18] defined a linear transformation $D_{\alpha \mathrm{,}\beta \mathrm{,}\gamma }^{s}f$ by

$D_{\alpha \mathrm{,}\beta \mathrm{,}\gamma }^{s}f\left(z\right)=z+{\displaystyle \underset{n=2}{\overset{\infty }{\sum }}}{\left(\frac{\alpha +n\beta +n^2\gamma }{\alpha +\beta +\gamma }\right)}^s{a}_n^i{z}^n\mathrm{,}\beta \mathrm{,}\gamma \ge \mathrm{0\; <mo>\; ;\; </mo>}\alpha \ge \mathrm{1\; <mo>\; ;\; </mo>}s\in \mathbb{N}\cup \mathrm{0,}i\left(1\le i\le k\right)\mathrm{.}$ (6)

where $k\in \mathbb{N}$.

Motivated by the work of Kanas and Darwish, using the subclass of Kumar et al, involving the linear transformation in Equation (6), we study the coefficient estimates and solved the Fekete-Szegö problem for the subclass $S_n\left(b\right)$ involving the linear transformation $D_{\alpha \mathrm{,}\beta \mathrm{,}\gamma }^s$.

Definition 2 Let b be a nonzero complex number, and f a univalent function of the form (1) such that $H^n\left(z\right)\ne 0$ for $z\in U-\left\{0\right\}$. We say that f belongs to $S_n\left(b\right)$ if

$\mathrm{Re}\left\{1+\frac{1}{b}\left(\frac{z{\left(H^n\right)}^{\prime }\left(z\right)}{H^n\left(z\right)}-1\right)\right\}>0,z\in U,$ (7)

where $H=D_{\alpha ,\beta ,\gamma }^{s}f$ is as given in Equation (6).

The following results shall be employed in the proof of the main results of this study.

Lemma 1 [19] Let P be the class of analytic functions in U with $p\left(0\right)=1$, Re $p\left(z\right)>0$ and of the form

$p\left(z\right)=1+c_{1}z+c_2{z}^2+\cdots ,$ (8)

then

$\left|c_n\right|\le \mathrm{2,}n\ge 1.$

If $\left|c_1\right|=2$, then $p\left(z\right)\equiv p_1=\frac{1+{\gamma }_{1}z}{1-{\gamma }_{1}z}$ with ${\gamma }_1=\frac{c_1}{2}$. Conversely, if $p\left(z\right)\equiv p_1$ for some ${\gamma }_1=1$, then $c_1=2{\gamma }_1$ and $\left|c_1\right|=2$. Furthermore, we have

$\left|c_2-\frac{c_1^2}{2}\right|\le 2-\frac{{\left|c_1\right|}^2}{2}\mathrm{.}$

If $\left|c_1\right|<2$ and $\left|c_2-\frac{c_1^2}{2}\right|=2-\frac{{\left|c_1\right|}^2}{2}$, then $p\left(z\right)\equiv p_2$, where

$p_2\left(z\right)=\frac{1+z\frac{{\gamma }_{2}z+{\gamma }_1}{1+{\gamma }_1{\gamma }_{2}z}}{1-z\frac{{\gamma }_{2}z+{\gamma }_1}{1+{\gamma }_1{\gamma }_{2}z}}$

and ${\gamma }_1=\frac{c_1}{2}$, ${\gamma }_2=\frac{2c_2-c_1^2}{4-{\left|c_1\right|}^2}$. Conversely, if $p\left(z\right)=p_2$ for some ${\gamma }_1<1$ and ${\gamma }_2=1$, then ${\gamma }_1=\frac{c_1}{2}$, ${\gamma }_2=\frac{2c_2-c_1^2}{4-{\left|c_1\right|}^2}$ and $\left|c_2-\frac{c_1^2}{2}\right|=2-\frac{{\left|c_1\right|}^2}{2}$.

In what follows, we give the statement and proof of the results of this study.

2. Coefficient Estimates for...

Theorem 1 Let $n\ge 0$ and b a non-zero complex number. If f of the form (1) is in $S_n\left(b\right)$, then

$\left|a_2^i\right|\le 2\left|b\right|{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s$

and

$\left|a_3^i\right|\le \left|b\right|{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\text{max}\left[\mathrm{1,}\left|1+2b\right|\right]\mathrm{,}\beta \mathrm{,}\gamma \ge \mathrm{0\; <mo>\; ;\; </mo>}\alpha \ge \mathrm{1\; <mo>\; ;\; </mo>}s\in \mathbb{N}\cup \mathrm{0,}i\left(1\le i\le k\right)\mathrm{.}$

Proof 1 Let $f\in S_n\left(b\right)$, then by definition 2, there exist a class of analytic function p given by

$p\left(z\right)=1+c_{1}z+c_2{z}^2+\cdots$

satisfying $P\left(0\right)=1$ and $\mathrm{Re}\left(p\left(z\right)\right)>0$ such that

$1+\frac{1}{b}\left(\frac{z{\left(H^n\right)}^{\prime }\left(z\right)}{H^n\left(z\right)}-1\right)=1+c_{1}z+c_2{z}^2+\cdots$ (9)

where $H=D_{\alpha \beta \gamma }^s.$

From Equation (9), we have:

$\frac{z{\left(H^n\right)}^{\prime }\left(z\right)}{H^n\left(z\right)}=1+b\left(p\left(z\right)-1\right)$ (10)

Equating coefficients in Equation (10) using Equation (6) with $D_{\alpha \mathrm{,}\beta \mathrm{,}\gamma }^{s}f\left(z\right)=z+A_2{z}^2+A_3{z}^3+\cdots$, we have

$A_2=b{c}_1$ (11)

$A_3=\frac{b}{2}\left[c_2+b{c}_1^2\right]$ (12)

$\equiv \frac{b}{2}\left(c_2-\frac{c_1^2}{2}\right)+\left(1+2b\right)\frac{b{c}_1^2}{4}$ (13)

From Equations (12) and (13) using Equation (6), we have,

$a_2=b{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s{c}_1$ (14)

respectively

$a_3=\frac{b}{2}{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\left[c_2+b{c}_1^2\right]$ (15)

On the account of Equations (14) and (15) using Lemma 1, we have

$\left|a_2\right|\mathrm{<mo>\; =\; </mo>}\left|b{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s{c}_1\right|\le 2\left|b\right|{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s$

and

$\begin{array}{c}\left|a_3\right|=\left|\frac{b}{2}{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\left[c_2-\frac{c_1}{2}+\frac{1+2b}{2}{c}_1^2\right]\right|\\ \le \frac{\left|b\right|}{2}{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\left[2-\frac{\left|c_1\right|}{2}+\frac{\left|1+2b\right|}{2}{\left|c_1\right|}^2\right]\\ =\frac{\left|b\right|}{2}{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\left[2+\frac{\left|c_1\right|}{2}\left(\left|1+2b\right|-1\right)\right]\\ \le \left|b\right|{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\left[\mathrm{1,1}+\left|1+2b\right|-1\right]\\ =\left|b\right|{\left(\frac{\alpha +\beta +\gamma }{\alpha +2\beta +4\gamma }\right)}^s\text{max}\left[\mathrm{1,}\left|1+2b\right|\right]\mathrm{.}\end{array}$

which proves theorem 1.

3. The Fekete-Szegö Problem for the Subclasses $S_n(b)$

Theorem 2 Let b be a nonzero complex number and $f\in S_n\left(b\right)$. Then $\mu \in C$, the following holds.

$\left|a_3-\mu a_2^2\right|\le b{\left(\frac{\alpha +\beta +\gamma }{\alpha +3\beta +9\gamma }\right)}^s\text{max}\left\{\mathrm{1,}\left|1+2b-2b\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right|\right\}$

Proof 2 From Equations (14) and (15), we have

$\begin{array}{c}{a}_3-\mu a_2^2=\frac{b}{2}{t}_3^{-s}\left[c_2+b{c}_1^2\right]-\mu {\left(b{t}_2^{-s}{c}_1\right)}^2\\ =\frac{b}{2}{t}_3^{-s}\left[c_2+b{c}_1^2-2b\mu t_3^{-s}{t}_2^{2s}{c}_1^2\right]\\ =\frac{b}{2}{t}_3^{-s}\left[c_2-\frac{c_1^2}{2}+\frac{c_1^2}{2}\left(1+2b-2b\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right)\right]\end{array}$

where $t_2=\left(\frac{\alpha +2\beta +4\gamma }{\alpha +\beta +\gamma }\right)$ and $t_3=\left(\frac{\alpha +3\beta +9\gamma }{\alpha +\beta +\gamma }\right)$.

Applying Lemma 1 to the above last inequality, we obtain

$\begin{array}{c}\left|a_3-\mu a_2^2\right|\le \frac{b}{2}{t}_3^{-s}\left[2+\frac{c_1^2}{2}\left(\left|1+2b-2b\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right|-1\right)\right]\\ \le b{t}_3^{-s}\text{max}\left\{\mathrm{1,}\left(\left|1+2b-2b\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right|\right)\right\}\end{array}$ (16)

This proves the theorem.

Theorem 3 Let b be a nonzero complex number and $f\in S_n\left(b\right)$. Then $\mu \in ℝ$, the following holds.

$\left|a_3-\mu a_2^2\right|\le (\begin{array}{ll}b{t}_3^{-s}\left[\left|1+2b\left(1-\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right)\right|\right]\hfill & \text{if}\mu \le t_3^{-s}\hfill \\ b{t}_3^{-s}\hfill & \text{if}{t}_3^{-s}\le \mu \le t_3^{-s}\frac{1+2b}{2b}\hfill \\ b{t}_3^{-s}\left[\left|2b\left(\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}-1\right)-1\right|\right]\hfill & \text{if}\mu \ge t_3^{-s}\frac{1+2b}{2b}\hfill \end{array}$ (17)

where $t_3=\left(\frac{\alpha +3\beta +9\gamma }{\alpha +\beta +\gamma }\right)$.

Proof 3 Let $\mu \le t_3^{-s}$. From Equation (17), we have

$\left|a_3-\mu a_2^2\right|\le b{t}_3^{-s}\left[\left|1+2b\left(1-\mu \frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}\right)\right|\right]$

Now, using the above calculations with $t_3^{-s}\le \mu \le t_3^{-s}\frac{1+2b}{2b}$, we have

$\left|a_3-\mu a_2^2\right|\le b{t}_3^{-s}$

and conclusively, let $\mu \ge t_3^{-s}\frac{1+2b}{2b}$, then

$\begin{array}{c}\left|a_3-\mu a_2^2\right|\le \frac{b}{2}{t}_3^{-s}\left[2+\frac{{\left|c_1\right|}^2}{2}\left(2\mu b\frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}-2-2b\right)\right]\\ \le b{t}_3^{-s}\left[2\mu b\frac{{\left(\alpha +3\beta +9\gamma \right)}^s}{{\left(\alpha +2\beta +4\gamma \right)}^{2s}}-1-2b\right]\end{array}$

This concludes the proof of the theorem 3.

4. Conclusions

The result in this paper extends the work of Kanas and Darwish as it is evident

that for $s=1$ and $\alpha +\beta +\gamma =\frac{\alpha +2\beta +4\gamma }{2}$, $s=0$ in the first part, respectively second part of the theorem 1 yields the first part respectively second part

of the theorem 2.2 for $n=0$ in Kanas and Darwish. Moreover when $n>1$ and $\beta ,\gamma \ge 0;\alpha \ge 1$ in Equation (6), the result in this study gives finer initial coefficient estimates and bound for Fekete-Szegö problem.

It will also be interesting to check the effect of the linear transformation given in (6) on other subclasses of normalized analytic functions.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Fekete, M. and Szegö, G. (1933) Eine Bemerkung ber ungerade schlichte Funktionen. Journal of the London Mathematical Society, 8, 85-89. https://doi.org/10.1112/jlms/s1-8.2.85
[2]	Kanasa, S. and Darwish, H.E. (2010) Fekete-Szegö Problem for Starlike and Convex Functions of Complex Order. Applied Mathematics Letters, 23, 777-782. https://doi.org/10.1016/j.aml.2010.03.008
[3]	Abdel-Gawad, H.R. and Thomas, D.K. (1992) The Fekete-Szegö Problem for Strongly Close-to-Convex Functions. Proceedings of the AMS, 114, 345-349. https://doi.org/10.2307/2159653
[4]	Al-Amiri, H.S. (1979) Certain Generalization of Prestarlike Functions. Journal of the Australian Mathematical Society, 28, 325-334. https://doi.org/10.1017/S1446788700012283
[5]	Choi, J.H., Kim, Y.C. and Sugawa, T. (2007) A General Approach to the Fekete-Szegö Problem. Journal of the Mathematical Society of Japan, 59, 707-727. https://doi.org/10.2969/jmsj/05930707
[6]	Chonweerayoot, A, Thomas, D.K. and Upakarnitikaset, W. (1992) On the Fekete-Szegö Theorem for Close-to-Convex Functions. Publications de l’Institut Mathématique, 66, 18-26.
[7]	Darus, M. and Thomas, D.K. (1996) On the Fekete-Szegö Theorem for Close-to-Convex Functions. Mathematica Japonica, 44, 507-511.
[8]	Keogh, F.R. and Merkes, E.P. (1969) A Coefficient Inequality for Certain Classes of Analytic Functions. Proceedings of the AMS, 20, 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
[9]	Koepf, W. (1987) On the Fekete-Szegö Problem for Close-to-Convex Functions. Proceedings of the AMS, 101, 89-95. https://doi.org/10.2307/2046556
[10]	Kanas, S. and Lecko, A. (1990) On the Fekete-Szegö Problem and the Domain Convexity for a Certain Class of Univalent Functions. Folia Sci. Univ. Tech. Resov., 73, 49-58.
[11]	London, R.R. (1993) Fekete-Szegö Inequalities for Close-to-Convex Functions. Proceedings of the AMS, 117, 947-950. https://doi.org/10.2307/2159520
[12]	Ma, W. and Minda, D. (1994) A Unified Treatment of Some Special Classes of Univalent Functions. In: Li, Z., Ren, F., Yang, L. and Zhang, S., Eds., Proceeding of Conference on Complex Analytic, International Press, New York, 157-169.
[13]	Aouf, M.K., El-Ashwah, R.M. and El-Deeb, S.M. (2014) Fekete-Szegö Inequalities for Starlike Functions with Respect to k-Symmetric Points of Complex Order, Journal of Complex Analysis, 2014, Article ID: 131475. https://doi.org/10.1155/2014/131475
[14]	Nasr M.A. and Aouf, M.K. (1985) Starlike Function of Complex Order. Journal of Natural Sciences and Mathematics, 25, 1-12.
[15]	Wiatrowski, P. (1971) The Coefficients of a Certain Family of Holomorphic Functions. Zeszyty Naukowe Uniwesytetu Lodzkiego, Seria: Nauki Matematyzno-Przyrodnicze, No. 2, 75-85.
[16]	Nasr M.A. and Aouf, M.K. (1982) On Convex Functions of Complex Order. Mansoura Science Bulletin, 9, 565-582.
[17]	Kumar, V., Shukla, S.L. and Chaudhary, A.M. (1990) On a Class of Certain Analytic Functions of Complex Order. Tamkang Journal of Mathematics, 21, 101-109.
[18]	Makinde, D.O., Hamzat, O. and Gbolagade, A.M. (2019) A Generalized Multiplier Transform on a Univalent Integral Operator. Journal of Contemporary Applied Mathematics, 9, 31-38.
[19]	Pommerenke, C. (1975) Univalent Functions. In: Studia Mathematica Mathematische Lehrbucher, Vandenhoeck and Ruprecht.