γ z 2 + c 3 cos γ z 3 + = cos γ + q 1 z + q 2 z 2 + q 3 z 3 + (3.3)

Comparing coefficients of (3.1) and (3.3) gives

${a}_{2}=\frac{{c}_{1}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma }{{2}^{n+1}}$ (3.4)

${a}_{3}=\frac{{c}_{2}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma +{\beta }^{2}{\text{e}}^{-2i\gamma }}{{3}^{n+1}}$ (3.5)

${a}_{4}=\frac{{c}_{3}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma +{c}_{1}{\beta }^{2}{\text{e}}^{-3i\gamma }\mathrm{cos}\gamma }{{4}^{n+1}}$ (3.6)

Solving for the bounds of (3.4), (3.5), (3.6) and using lemma 2.1 give

$|{a}_{2}|\le \frac{\mathrm{cos}\gamma }{{2}^{n}}$ (3.7)

$|{a}_{3}|\le \frac{2\mathrm{cos}\gamma +{\beta }^{2}}{{3}^{n+1}}$ (3.8)

$|{a}_{4}|\le \frac{2\mathrm{cos}\gamma +2{\beta }^{2}\mathrm{cos}\gamma }{{4}^{n+1}}$ (3.9)

Remark 2

For $n=0$

$|{a}_{2}|\le \mathrm{cos}\gamma$

$|{a}_{3}|\le \frac{2\mathrm{cos}\gamma +{\beta }^{2}}{3}$

Theorem 3.2. Let $f\left(z\right)\in {C}_{n}\left(\beta ,\gamma \right)$, then for any real number $\mu$

$|{a}_{3}-\mu {a}_{2}^{2}|\le \left\{\begin{array}{ll}\frac{{\beta }^{2}+2\mathrm{cos}\gamma }{{3}^{n+1}}-\frac{\mu {\text{e}}^{-i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2n}}\hfill & \text{if}\text{\hspace{0.17em}}\mu \le 0\hfill \\ \frac{{\beta }^{2}+2\mathrm{cos}\gamma }{{3}^{n+1}}\hfill & \text{if}\text{\hspace{0.17em}}0\le \mu \le \frac{{2}^{2n+2}}{{3}^{n+1}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma }\hfill \\ \frac{{\beta }^{2}-2\mathrm{cos}\gamma }{{3}^{n+1}}+\frac{\mu {\text{e}}^{-i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2n}}\hfill & \text{if}\text{\hspace{0.17em}}\mu \ge \frac{{2}^{2n+2}}{{3}^{n+1}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma }\hfill \end{array}$

Proof:

Using (3.4) and (3.5) give

$\begin{array}{c}|{a}_{3}-\mu {a}_{2}^{2}|=|\frac{{c}_{2}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma }{{3}^{n+1}}+\frac{{\beta }^{2}{\text{e}}^{-2i\gamma }}{{3}^{n+1}}-\frac{\mu {c}_{1}^{2}{\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2n+2}}|\\ \le \frac{{\beta }^{2}}{{3}^{n+1}}+\frac{\mathrm{cos}\gamma }{{3}^{n+1}}|{c}_{2}-\frac{{3}^{n+1}\mu {\text{e}}^{-i\gamma }\mathrm{cos}\gamma }{{2}^{2n+1}}\frac{{c}_{1}^{2}}{2}|\end{array}$ (3.10)

then using lemma (2.2) in (3.10) gives

$|{a}_{3}-\mu {a}_{2}^{2}|\le \frac{{\beta }^{2}+2\mathrm{cos}\gamma }{{3}^{n+1}}-\frac{\mu {\text{e}}^{-i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2n}}$ (3.11)

Let

$0\le \frac{{3}^{n+1}\mu {\text{e}}^{-i\gamma }\mathrm{cos}\gamma }{{2}^{2n+1}}\le 2$

then by lemma 2.2 we obtain

$|{a}_{3}-\mu {a}_{2}^{2}|\le \frac{{\beta }^{2}+2\mathrm{cos}\gamma }{{3}^{n+1}}$ (3.12)

suppose

$\frac{{3}^{n+1}\mu {\text{e}}^{-i\gamma }}{{2}^{2n+1}}\ge 2$

then using lemma 2.2 gives

$|{a}_{3}-\mu {a}_{2}^{2}|\le \frac{{\beta }^{2}-2\mathrm{cos}\gamma }{{3}^{n+1}}+\frac{\mu {\text{e}}^{-i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2n}}$ (3.13)

Theorem 3.3 Let $f\left(z\right)\in {C}_{n}\left(\beta ,\gamma \right),\beta \in \left[0,1\right],\gamma \in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$ and $n\in {ℕ}_{0}$

then

${H}_{2}\left(2\right)=|{a}_{2}{a}_{4}-{a}_{3}^{2}|\le \frac{{\beta }^{4}+4{\beta }^{2}\mathrm{cos}\gamma +4{\mathrm{cos}}^{2}\gamma }{{3}^{2n+2}}+\frac{\left({\beta }^{4}+6{\beta }^{2}+9\right)\mathrm{cos}\gamma }{{2}^{3n+4}}$

Proof:

Using (3.4), (3.5) and (3.6) give

$\begin{array}{l}|{a}_{2}{a}_{4}-{a}_{3}^{2}|\\ =|\frac{{c}_{1}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma }{{2}^{n+1}}\left(\frac{{c}_{3}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma +{\beta }^{2}{c}_{1}{\text{e}}^{-3i\gamma }\mathrm{cos}\gamma }{{4}^{n+1}}\right)-{\left(\frac{{c}_{2}{\text{e}}^{-i\gamma }\mathrm{cos}\gamma +{\beta }^{2}{\text{e}}^{-2i\gamma }}{{3}^{n+1}}\right)}^{2}|\end{array}$ (3.14)

$\begin{array}{l}|{a}_{2}{a}_{4}-{a}_{3}^{2}|\\ =|\frac{{c}_{1}^{4}{\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}+\frac{{c}_{1}^{2}\left(4-{c}_{1}^{2}\right){\text{e}}^{-2i\gamma }x{\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}-\frac{{c}_{1}^{2}\left(4-{c}_{1}^{2}\right){\text{e}}^{-2i\gamma }{x}^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{c}_{1}\left(4-{c}_{1}^{2}\right)\left(1-{|x|}^{2}\right){\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma z}{{2}^{3n+4}}+\frac{{c}_{1}^{2}{\beta }^{2}{\text{e}}^{-4i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{3n+3}}-\frac{{c}_{1}^{4}{\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2\left(n+1\right)}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{x\left(4-{c}_{1}^{2}\right){c}_{1}^{2}{\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma }{2\cdot {3}^{2n+2}}-\frac{{c}_{1}^{2}{\alpha }^{2}{\text{e}}^{-3i\gamma }\mathrm{cos}\gamma }{{3}^{2n+2}}-\frac{{x}^{2}{\left(4-{c}_{1}^{2}\right)}^{2}{\text{e}}^{-2i\gamma }{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\beta }^{2}x\left(4-{c}_{1}^{2}\right){\text{e}}^{-3i\gamma }\mathrm{cos}\gamma }{{3}^{2n+2}}-\frac{{\beta }^{4}{\text{e}}^{-4i\gamma }}{{3}^{2n+2}}|\end{array}$ (3.15)

Suppose ${c}_{1}=c$, and recall that $|{c}_{1}|\le 2$, and assuming without restriction that $c\in \left[0,2\right]$. Then, using triangle inequality

(3.15) becomes

$\begin{array}{c}|{a}_{2}{a}_{4}-{a}_{3}^{2}|\le \frac{{c}^{4}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}+\frac{{c}^{2}\left(4-{c}^{2}\right)|x|{\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{{c}^{2}\left(4-{c}^{2}\right){|x|}^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{c\left(4-{c}^{2}\right)\left(1-{|x|}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{{c}^{2}{\beta }^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+3}}+\frac{{c}^{4}{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{|x|\left(4-{c}^{2}\right){c}^{2}{\mathrm{cos}}^{2}\gamma }{2\cdot {3}^{2n+2}}+\frac{{|x|}^{2}{\left(4-{c}^{2}\right)}^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{c}^{2}{\beta }^{2}\mathrm{cos}\gamma }{{3}^{2n+2}}+\frac{{\beta }^{2}|x|\left(4-{c}^{2}\right)\mathrm{cos}\gamma }{{3}^{2n+2}}+\frac{{\beta }^{4}}{{3}^{2n+2}}\end{array}$ (3.16)

Now, putting $\psi =|x|\le 1$ then

$\begin{array}{l}|{a}_{2}{a}_{4}-{a}_{3}^{2}|\\ \le \left\{\frac{{c}^{4}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}+\frac{c\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{{c}^{2}{\beta }^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{3n+3}}+\frac{{c}^{4}{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}+\frac{{c}^{2}{\beta }^{2}\mathrm{cos}\gamma }{{3}^{2n+2}}+\frac{{\beta }^{4}}{{3}^{2n+2}}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left\{\frac{{c}^{2}\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{\left(4-{c}^{2}\right){c}^{2}{\mathrm{cos}}^{2}\gamma }{2\cdot {3}^{2n+2}}+\frac{{\beta }^{2}\left(4-{c}^{2}\right)\mathrm{cos}\gamma }{{3}^{2n+2}}\right\}\psi \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left\{\frac{{c}^{2}\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}-\frac{c\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{{\left(4-{c}^{2}\right)}^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}\right\}{\psi }^{2}=F\left(c,\psi \right)\end{array}$ (3.17)

Differentiating $F\left(c,\psi \right)$ partially with respect to $\psi$ in the closed interval $0\le \psi \le 1$

$\begin{array}{c}\frac{\partial F\left(c,\psi \right)}{\partial \psi }=\left\{\frac{{c}^{2}\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{\left(4-{c}^{2}\right){c}^{2}{\mathrm{cos}}^{2}\gamma }{2\cdot {3}^{2n+2}}+\frac{{\beta }^{2}\left(4-{c}^{2}\right)\mathrm{cos}\gamma }{{3}^{2n+2}}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left\{\frac{{c}^{2}\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+5}}-\frac{c\left(4-{c}^{2}\right){\mathrm{cos}}^{2}\gamma }{{2}^{3n+4}}+\frac{{\left(4-{c}^{2}\right)}^{2}{\mathrm{cos}}^{2}\gamma }{{2}^{2}\cdot {3}^{2n+2}}\right\}\psi \\ >0\end{array}$ (3.18)

for $0\le \psi \le 1$, therefore is an increasing function. Hence, it attains maximum point at. Thus,

(3.19)

Now, the critical points occur at

but the maximum point occurring at [3.19] becomes

(3.20)

Therefore,

( The Fekete Szegö Functional and Second Hankel Determinant for a Certain Sublass of Analytic Functions

The Fekete Szegö Functional and Second Hankel Determinant for a Certain Sublass of Analytic Functions

Let S denote the class of functions that are analytic, normalized and univalent in the open unit disk E = {z: |z| <1}. Subclasses of S are the class of starlike and convex functions denoted by S* and C respectively. A new subclass of analytic functions that generalize some known subclasses of analytic functions was defined and investigated. We obtained coefficient bounds, upper estimates for the Fekete-Szegö functional and the Hankel determinant.

Cite this paper

Ayinla, R. and Opoola, T. (2019) The Fekete Szegö Functional and Second Hankel Determinant for a Certain Sublass of Analytic Functions. Applied Mathematics, 10, 1071-1078. doi: 10.4236/am.2019.1012074.

1. Introduction

Let A denote the class of functions

$f\left(z\right)=z+{a}_{2}{z}^{2}+{a}_{3}{z}^{3}+{a}_{4}{z}^{4}+\cdots$ (1.1)

which are analytic in the open unit disk $U=\left\{z:|z|<1\right\}$ and satisfy the condition $f\left(0\right)=0$ and ${f}^{\prime }\left(0\right)=1$.

Let S denote the subclass of A consisting of univalent in U. A function $f\left(z\right)\in S$ is said to be starlike in the unit disk if and only if

$Re\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}>0,z\in U$ (1.2)

Also, a function $f\left(z\right)\in S$ is said to be convex in the unit disk if and only if

$Re\left(1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right)>0,z\in U$ (1.3)

Let ${D}^{n}:A\to A$ be defined by

${D}^{0}f\left(z\right)=f\left(z\right)$

${D}^{1}f\left(z\right)=z{f}^{\prime }\left(z\right)$

${D}^{n}f\left(z\right)=z{\left[{D}^{n-1}f\left(z\right)\right]}^{\prime }$

which is equivalent to

${D}^{n}f\left(z\right)=z+\underset{k=2}{\overset{\infty }{\sum }}\text{ }\text{ }{k}^{n}{a}_{k}{z}^{k},\left(n=\left\{0,1,2,\cdots \right\}\right),z\in U$

${D}^{n}$ is the Salagean differential operator [1].

Fekete and Szegö [2] studied the estimate of a functional $|{a}_{3}-\sigma {a}_{2}^{2}|$ known as Fekete-Szegö functional, where $\sigma$ is real. Also, Noonan and Thomas [3] defined the qth Hankel determinant of $f\left(z\right)$ for $q\ge 1,n\ge 0$ by

${H}_{q}\left(n\right)=|\begin{array}{cccc}{a}_{n}& {a}_{n+1}& \cdots & {a}_{n+q-1}\\ {a}_{n+1}& {a}_{n+2}& \cdots & {a}_{n+q}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n+q-1}& {a}_{n+q}& \cdots & {a}_{n+2q-2}\end{array}|\left({a}_{1}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}f\left(z\right)\in S\right)$

This determinant has been considered for specific values q and n by many authors. It is well established that the Fekete-Szegö functional given by $|{a}_{3}-{a}_{2}^{2}|={H}_{2}\left(1\right)$. Pommerenke [4] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as starlike functions. Noor [5] investigated the Hankel determinant problem for the class of functions with bounded boundary rotation. Janteng et al. [6] studied the sharp upper bound for second Hankel determinant ${H}_{2}\left(2\right)=|{a}_{2}{a}_{4}-{a}_{3}^{2}|$ for univalent functions whose derivative has positive real parts. Also, Lee et al. [7] obtained bounds on second Hankel determinants belonging to the subclasses of Ma-Minda starlike and convex functions. Bansal [8] has obtained bounds on ${H}_{2}\left(2\right)$ for a new class of analytic functions.

In this paper, we obtained the coefficient bound, Fekete-Szegö functional and second Hankel determinant for the functions belonging to the subclass ${C}_{n}\left(\beta ,\gamma \right)$.

Definition 1.1. A function $f\left(z\right)$ of the form (1.1) analytic and univalent in U is said to be in the ${C}_{n}\left(\beta ,\gamma \right),\beta \in \left[0,1\right],\gamma \in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$ and $n\in {ℕ}_{0}$ if it satisfies the inequality

$Re\left\{{\text{e}}^{i\gamma }\left(1-{\text{e}}^{-2i\gamma }{\beta }^{2}{z}^{2}\right)\frac{{D}^{n+1}f\left(z\right)}{z}\right\}>0,\text{\hspace{0.17em}}z\in U.$ (1.4)

Remark 1

(1) For $n=0,\beta =0$ the class ${C}_{0}\left(0,\gamma \right)$ gives

$Re\left\{{\text{e}}^{i\gamma }{f}^{\prime }\left(z\right)\right\}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}z\in U$ (1.5)

studied in [9].

(2) For $n=0,\gamma =0$ gives

$Re\left\{\left(1-{\beta }^{2}{z}^{2}\right){f}^{\prime }\left(z\right)\right\}>0,z\in U.$ (1.6)

investigated by [10].

For $n=0$, the class gives

$Re\left\{{\text{e}}^{i\gamma }\left(1-{\text{e}}^{-2i\gamma }{\beta }^{2}{z}^{2}\right){f}^{\prime }\left(z\right)\right\}>0,z\in U.$ (1.7)

studied in [11].

2. Preliminary Lemmas

We need the following lemmas to prove our results.

Let P denote the class of Caratheodory functions.

$p\left(z\right)=1+{c}_{1}z+{c}_{2}{z}^{2}+{c}_{3}{z}^{3}+\cdots \left(z\in U\right)$

which are analytic and satisfy $p\left(0\right)=1$ and $\Re p\left(z\right)>0$

Lemma 2.1. Let $p\in P$. Then

$|{c}_{k}|\le 2\left(k\in ℕ\right)$ [12] (2.1)

Lemma 2.2. Let $p\in P$, then for any real $\lambda$

$|{c}_{2}-\lambda \frac{{c}_{1}^{2}}{2}|\le \left\{\begin{array}{ll}2\left(1-\lambda \right)\hfill & \text{if}\text{\hspace{0.17em}}\text{ }\lambda \le 0\hfill \\ 2\hfill & \text{if}\text{\hspace{0.17em}}\text{ }0\le \lambda \le 2\hfill \\ 2\left(\lambda -1\right)\hfill & \text{if}\text{\hspace{0.17em}}\text{ }\lambda \ge 2\hfill \end{array}$ [13] (2.2)

Lemma 2.3. Let $p\in P$ then

$2{c}_{2}={c}_{1}^{2}+x\left(4-{c}_{1}^{2}\right)$ (2.3)

$4{c}_{3}={c}_{1}^{3}+2{c}_{1}\left(4-{c}_{1}^{2}\right)x-{c}_{1}\left(4-{c}_{1}^{2}\right){x}^{2}+2\left(4-{c}_{1}^{2}\right)\left(1-{|x|}^{2}\right)z$ (2.4)

for some value of $x,z$, such that $|x|\le 1$ and $|z|\le 1$ [14].

3. Main Results

Theorem 3.1. Let $f\left(z\right)\in {C}_{n}\left(\beta ,\gamma \right),\beta \in \left[0,1\right],\gamma \in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$ and $n\in {ℕ}_{0}$.

Then

$|{a}_{2}|\le \frac{\mathrm{cos}\gamma }{{2}^{n}}$

$|{a}_{3}|\le \frac{2\mathrm{cos}\gamma +{\beta }^{2}}{{3}^{n+1}}$

Proof:

Let $f\left(z\right)\in {C}_{n}\left(\beta ,\gamma \right)$, then by [1.4]

$Re{\text{e}}^{i\gamma }\left[\left(1-{\text{e}}^{-2i\gamma }{\beta }^{2}{z}^{2}\right)\frac{{D}^{n+1}f\left(z\right)}{z}\right]>0,\gamma \in \left(\frac{-\pi }{2},\frac{\pi }{2}\right),0\le \beta \le 1,n\in {ℕ}_{0},z\in U$

Now,

$\begin{array}{l}{\text{e}}^{i\gamma }\left[\left(1-{\text{e}}^{-2i\gamma }{\beta }^{2}{z}^{2}\right)\frac{{D}^{n+1}f\left(z\right)}{z}\right]\\ ={\text{e}}^{i\gamma }+{q}_{1}z+{q}_{2}{z}^{2}+\cdots =\left(\mathrm{cos}\gamma +i\mathrm{sin}\gamma \right)+\underset{n=1}{\overset{\infty }{\sum }}\text{ }\text{ }{q}_{n}{z}^{n}\end{array}$ (3.1)

Then

$\exists q\left(z\right)=\mathrm{cos}\gamma +i\mathrm{sin}\gamma +\underset{n=1}{\overset{\infty }{\sum }}\text{ }\text{ }{q}_{n}{z}^{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}z\in U,n\in ℕ$

${\text{e}}^{i\gamma }\left[\left(1-{\text{e}}^{-2i\gamma }{\beta }^{2}{z}^{2}\right)\frac{{D}^{n+1}f\left(z\right)}{z}\right]=p\left(z\right)\mathrm{cos}\gamma +i\mathrm{sin}\gamma$ (3.2)

that is