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 .

Fekete and Szegö  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  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  investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as starlike functions. Noor  investigated the Hankel determinant problem for the class of functions with bounded boundary rotation. Janteng et al.  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.  obtained bounds on second Hankel determinants belonging to the subclasses of Ma-Minda starlike and convex functions. Bansal  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 .

(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 .

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 .

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

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

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

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

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