On a Fekete–Szegö Problem Associated with Generalized Telephone Numbers

Abstract

One of the important problems regarding coefficients of analytical functions (i.e., Fekete–Szegö inequality) was raised by Fekete and Szegö in 1933. The results of this research are dedicated to determine upper coefficient estimates and the Fekete–Szegö problem in the class ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$, which is defined by generalized telephone numbers. We also indicate some specific conditions and consequences of results found by us.

1. Introduction

Denote by $\mathcal{A}$ the class of holomorphic functions in the open unit disk $U=\{z\in C:|z|<1\},$ of the style

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k.$$

(1)

We present by S the subclass of $\mathcal{A}$ occurring of functions, which are also univalent in U.

We say that $f\in S$ is known starlike of order $\gamma (0\leqq \gamma <1)$ if

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

and $f\in S$ is known convex of order $\gamma (0\leqq \gamma <1)$ if

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

We know that ${\mathcal{S}}^{\ast }\left(\gamma \right)$ and $\mathcal{C}\left(\gamma \right)$ are the classes of functions that are starlike with order $\gamma$ and convex with order $\gamma$ in the unit disk U.

The image of U under each univalent function $f\in \mathcal{A}$ contains a disk of radius $\frac{1}{4}$, see the Koebe $\frac{1}{4}$ theorem [1] and each function $f\in S$ has an inverse $f^{-1}$ defined by $f^{-1}\left(f\left(z\right)\right)=z$ and

$$f\left(f^{-1}\left(w\right)\right)=w,\left(\left|w\right|<r_0\left(f\right),r_0\left(f\right)\geqq \frac{1}{4}\right)$$

where

$$g\left(w\right)=f^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

$f\in \mathcal{A}$ is known to be a bi-univalent function if each of functions f and $f^{-1}$ are univalent functions in U. All bi-univalent functions belonging to U were indicated by $\mathsf{\Sigma }$.

Many works on the bi-univalent functions have been presented in the previous papers (see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]). We recall some examples of functions in the family $\mathsf{\Sigma },$ from the work of Srivastava et al. [21],

$$\frac{z}{1-z},-log(1-z)\mathrm{and}\frac{1}{2}log\left(\frac{1+z}{1-z}\right).$$

The history of the Fekete–Szegö question $\left|a_3-\eta a_2^2\right|$ for $f\in S$ has been well-known in the Geometric Function Theory. its origin lies in the refutation of the Little Wood-Paley conjecture by Fekete and Szegö (see [22]). It was denoted that the coefficients of univalent functions are limited to unity for this conjection. The Fekete–Szegö inequalities were found by many authors for different function families. This subject has attracted great interest among investigators in Geometric Function Theory. (see, for example, [4,6,16,17,21,23,24,25,26,27,28,29,30,31]).

Traditional phone numbers are enumerated by recurrence relationship

$$T\left(k\right)=T(k-1)+(k-1)T(k-2)k\geqq 2,$$

within initial terms

$$T\left(0\right)=T\left(1\right)=1.$$

For integers $k\geqq 0$ and $\tau \geqq 1$, Wloch and Wolowiec-Musial [32] defined generalized telephone numbers $T(\tau ,k)$ by the recurrence relation:

$$T(\tau ,k)=\tau T(\tau ,k-1)+(k-1)T(\tau ,k-2),$$

with initial conditions

$$T(\tau ,0)=1\mathrm{and}T(\tau ,1)=\tau .$$

Recently, Bednarz and Wolowiec-Musial [33] considered the accessible generalization of telephone numbers by

$$T\tau \left(k\right)=T\tau (k-1)+\tau (k-1)T\tau (k-2),$$

where $k\geqq 2$ and $\tau \geqq 1$ with initial conditions

$$T\tau \left(0\right)=T\tau \left(1\right)=1.$$

Very recently, Deniz [27] investigated the exponential-generating function for $T\tau \left(k\right)$ as follows:

$$e^{\left(r+\tau \frac{r}{2}\right)}=\sum _{k=0}^{\infty }T\tau \left(k\right)\frac{r^k}{k!}.$$

Clearly, when $\tau =1$, we have $T\tau \left(k\right)\equiv T\left(k\right)$ classical telephone numbers.

We research the following function:

$$\vartheta \left(z\right)=e^{\left(z+\tau \frac{z^2}{2}\right)}=1+z+\frac{z^2}{2}+\frac{1+\tau }{6}{z}^3+\frac{1+3\tau }{24}{z}^4+\cdots$$

(2)

with its domain of definition in U. It is worth noting that $\vartheta \left(z\right)$ is a holomorphic function in $U,$ with a positive real part, where $\vartheta \left(0\right)=1,{\vartheta }^{\prime }\left(0\right)>0$ and where $\vartheta$ maps U onto a starlike region according to 1 and symmetric according to the real axis.

Lemma 1 

([1], p. 41). Let $h\in \mathcal{P}$ be presented as $h\left(z\right)$, given below:

$$h\left(z\right)=1+c_{1}z+c_2{z}^2+\cdots ,wherez\in U.$$

The inequality given below confirms the sharp estimate

$$|c_n|\leqq 2,wheren\in \mathbb{N}.$$

2. Main Results

We now provide, using the generalized telephone numbers, the following subfamily of holomorphic and bi-univalent functions.

Definition 1. 

The family ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ contains all the functions $f\in \mathsf{\Sigma }$ if it fulfills the next subordinations:

$$(1-\delta )\left[(1-\lambda )\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+\lambda \left(1+\frac{z{f}^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)\right]+\delta \frac{\lambda z^2{f}^{\prime \prime }\left(z\right)+z{f}^{\prime }\left(z\right)}{\lambda z{f}^{\prime }\left(z\right)+(1-\lambda )f\left(z\right)}\prec e^{\left(z+\tau \frac{z^2}{2}\right)}=:\vartheta \left(z\right)$$

and

$$(1-\delta )\left[(1-\lambda )\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}+\lambda \left(1+\frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}\right)\right]+\delta \frac{\lambda w^2{g}^{\prime \prime }\left(w\right)+w{g}^{\prime }\left(w\right)}{\lambda w{g}^{\prime }\left(w\right)+(1-\lambda )g\left(w\right)}\prec e^{\left(w+\tau \frac{w^2}{2}\right)}=:\vartheta \left(w\right),$$

where $\delta \geqq 0$, $0\leqq \lambda \leqq 1$, $1\leqq \tau <2$ and $g\left(w\right)=f^{-1}\left(w\right)$.

Remark 1. 

1. If we take $\delta =\lambda =0$ in Definition 1, the class ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ turns into the class ${\mathcal{S}}_{\mathsf{\Sigma }}^{\ast }\left(\vartheta \right)$, which was studied recently by Cotîrlǎ and Wanas (see [34]).

2. If we take $\delta =0$ and $\lambda =1$ in Definition 1, the class ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ turns into the class ${\mathcal{C}}_{\mathsf{\Sigma }}\left(\vartheta \right)$, which was introduced recently by Cotîrlǎ and Wanas (see [34]).

Theorem 1. 

If f presented by (1) is in ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$$(\delta \geqq 0,0\leqq \lambda \leqq 1)$, then

$$|a_2|\leqq min\left\{\frac{1}{\lambda +1},\sqrt{\frac{2}{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}}\right\}$$

and

$$|a_3|\leqq min\left\{\frac{1}{2(2\lambda +1)}+\frac{\tau +1}{2\left(\lambda +1-\lambda \delta (\lambda -1)\right)},\frac{1}{{\left(\lambda +1\right)}^2}+\frac{1}{2(2\lambda +1)}\right\}.$$

Proof. 

Let $f\in {\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ and $f^{-1}=g$. There are the functions $\mathsf{\Phi },\mathsf{\Psi }:U⟶U$ holomorphic, with $\mathsf{\Phi }\left(0\right)=\mathsf{\Psi }\left(0\right)=0$, fulfills the equalities given below:

$$(1-\delta )\left[(1-\lambda )\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+\lambda \left(1+\frac{z{f}^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)\right]+\delta \frac{\lambda z^2{f}^{\prime \prime }\left(z\right)+z{f}^{\prime }\left(z\right)}{\lambda z{f}^{\prime }\left(z\right)+(1-\lambda )f\left(z\right)}=\vartheta \left(\mathsf{\Phi }\left(z\right)\right),z\in U$$

(3)

and

$$(1-\delta )\left[(1-\lambda )\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}+\lambda \left(1+\frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}\right)\right]+\delta \frac{\lambda w^2{g}^{\prime \prime }\left(w\right)+w{g}^{\prime }\left(w\right)}{\lambda w{g}^{\prime }\left(w\right)+(1-\lambda )g\left(w\right)}=\vartheta \left(\mathsf{\Psi }\left(w\right)\right),w\in U.$$

(4)

Define the functions x and y by

$$x\left(z\right)=\frac{1+\mathsf{\Phi }\left(z\right)}{1-\mathsf{\Phi }\left(z\right)}=1+x_{1}z+x_2{z}^2+\cdots$$

and

$$y\left(z\right)=\frac{1+\mathsf{\Psi }\left(z\right)}{1-\mathsf{\Psi }\left(z\right)}=1+y_{1}z+y_2{z}^2+\cdots .$$

It follows that $x,$y are analytic functions in U, where $x\left(0\right)=1=y\left(0\right)$. Then, we obtain $\mathsf{\Phi },\mathsf{\Psi }:U⟶U$, where x and y are the functions with a positive real part in U.

But, we have

$$\mathsf{\Phi }\left(z\right)=-\frac{1-x\left(z\right)}{x\left(z\right)+1}=\frac{1}{2}\left[x_{1}z+\left(x_2-\frac{x_1^2}{2}\right)z^2\right]+\cdots ,z\in U$$

(5)

and

$$\mathsf{\Psi }\left(z\right)=-\frac{1-y\left(z\right)}{y\left(z\right)+1}=\frac{1}{2}\left[y_{1}z+\left(y_2-\frac{y_1^2}{2}\right)z^2\right]+\cdots ,z\in U.$$

(6)

By taking the place of (5) and (6) into (3) and (4), using (2), we obtain

$$\begin{array}{c}(1-\delta )\left[(1-\lambda )\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}+\lambda \left(1+\frac{z{f}^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\right)\right]+\delta \frac{\lambda z^2{f}^{\prime \prime }\left(z\right)+z{f}^{\prime }\left(z\right)}{\lambda z{f}^{\prime }\left(z\right)+(1-\lambda )f\left(z\right)}\\ =\vartheta \left(\mathsf{\Phi }\left(z\right)\right)=e^{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}+\tau \frac{{\left(\frac{x\left(z\right)-1}{x\left(z\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{x}_{1}z+\left[\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8}\right]z^2+\cdots \end{array}$$

(7)

and

$$\begin{array}{c}(1-\delta )\left[(1-\lambda )\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}+\lambda \left(1+\frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}\right)\right]+\delta \frac{\lambda w^2{g}^{\prime \prime }\left(w\right)+w{g}^{\prime }\left(w\right)}{\lambda w{g}^{\prime }\left(w\right)+(1-\lambda )g\left(w\right)}\\ =\vartheta \left(\mathsf{\Psi }\left(w\right)\right)=e^{\left(\frac{y\left(w\right)-1}{1+y\left(w\right)}+\tau \frac{{\left(\frac{y\left(w\right)-1}{y\left(w\right)+1}\right)}^2}{2}\right)}=1+\frac{1}{2}{y}_{1}w+\left[\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}\right]w^2+\cdots \end{array}$$

(8)

Equating the coefficients in (7) and (8) yields

$$(\lambda +1)a_2=\frac{1}{2}{x}_1,$$

(9)

$$2(2\lambda +1)a_3-\left(\lambda \delta (\lambda -1)+3\lambda +1\right)a_2^2=\frac{x_2}{2}+\frac{(\tau -1)x_1^2}{8},$$

(10)

$$-(\lambda +1)a_2=\frac{1}{2}{y}_1$$

(11)

and

$$2(2\lambda +1)(2a_2^2-a_3)-\left(\lambda \delta (\lambda -1)+3\lambda +1\right)a_2^2=\frac{y_2}{2}+\frac{(\tau -1)y_1^2}{8}.$$

(12)

From (9) and (11), we have

$$x_1=-y_1$$

(13)

and

$$2{\left(\lambda +1\right)}^2{a}_2^2=\frac{1}{4}(x_1^2+y_1^2).$$

(14)

If we add (10) to (12), we obtain

$$2\left(\lambda +1-\lambda \delta (\lambda -1)\right)a_2^2=\frac{1}{2}(x_2+y_2)+\frac{1}{8}(\tau -1)(x_1^2+y_1^2).$$

(15)

Substituting from (14) the value of $x_1^2+y_1^2$ in the relation (15), we obtain

$$a_2^2=\frac{x_2+y_2}{2\left[2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right]}.$$

(16)

Applying Lemma 1 for the coefficients $x_1,x_2,y_1,y_2$ in (14) and (16), we obtain

$$|a_2|\leqq \frac{1}{\lambda +1},\left|a_2\right|\leqq \sqrt{\frac{2}{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}}.$$

To obtain $|a_3|$, from (10) we subtract (12) and using (13), we obtain $x_1^2=y_1^2$, hence

$$4(2\lambda +1)(a_3-a_2^2)=\frac{1}{2}(x_2-y_2),$$

(17)

then by substituting of the value of $a_2^2$ from (14) into (17), we obtain

$$a_3=\frac{x_1^2+y_1^2}{8{\left(\lambda +1\right)}^2}+\frac{x_2-y_2}{8(2\lambda +1)}.$$

So, we have

$$|a_3|\leqq \frac{1}{{\left(\lambda +1\right)}^2}+\frac{1}{2(2\lambda +1)}.$$

Also, substituting the value of $a_2^2$ from (15) into (17), we obtain

$$a_3=\frac{x_2-y_2}{8(2\lambda +1)}+\frac{x_2+y_2}{4\left(\lambda +1-\lambda \delta (\lambda -1)\right)}+\frac{(\tau -1)(x_1^2+y_1^2)}{16\left(\lambda +1-\lambda \delta (\lambda -1)\right)}$$

and we have

$$|a_3|\leqq \frac{1}{2(2\lambda +1)}+\frac{\tau +1}{2\left(\lambda +1-\lambda \delta (\lambda -1)\right)}.$$

□

When $\delta =\lambda =0$, the Theorem 1 is reduced to the corresponding results of Cotîrlǎ and Wanas (see [34]).

Corollary 1 

([34]). If f presented by (1) is in the class ${\mathcal{S}}_{\mathsf{\Sigma }}^{\ast }\left(\vartheta \right)$, then

$$|a_2|\leqq min\left\{1,\sqrt{\frac{2}{\left|3-\tau \right|}}\right\}$$

and

$$|a_3|\leqq min\left\{\frac{2+\tau }{2},\frac{3}{2}\right\}.$$

If we put $\delta =0$ and $\lambda =1$ in Theorem 1, the results reduced to the corresponding results of Cotîrlǎ and Wanas (see [34]).

Corollary 2 

([34]). Let f presented by (1) be in the family ${\mathcal{C}}_{\mathsf{\Sigma }}\left(\vartheta \right)$. Then,

$$|a_2|\leqq min\left\{\frac{1}{4},\sqrt{\frac{1}{2\left|2-\tau \right|}}\right\}$$

and

$$|a_3|\leqq min\left\{\frac{3\tau +5}{12},\frac{5}{12}\right\}.$$

Following theorem gives us inequalities of Fekete–Szegö for the class ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$.

Theorem 2. 

For $\delta \geqq 0$, $0\leqq \lambda \leqq 1$ and $\eta \in \mathbb{R}$, let $f\in {\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ be of the form (1). Then,

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2(2\lambda +1)};\left|\eta -1\right|\leqq \frac{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}{4(2\lambda +1)},\hfill \\ \\ \frac{2\left|\eta -1\right|}{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|};\left|\eta -1\right|\geqq \frac{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}{4(2\lambda +1)}.\hfill \end{array}\right.$$

Proof. 

It follows from (16) and (17) that

$$\begin{array}{cc}\hfill a_3-\eta a_2^2& =\frac{x_2-y_2}{8(2\lambda +1)}+\left(1-\eta \right)a_2^2\hfill \\ \hfill & =\frac{x_2-y_2}{8(2\lambda +1)}+\frac{(x_2+y_2)\left(1-\eta \right)}{2\left[2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right]}\hfill \\ \hfill & =\frac{1}{2}\left[\left(\psi (\eta ,\tau )+\frac{1}{4(2\lambda +1)}\right)x_2+\left(\psi (\eta ,\tau )-\frac{1}{4(2\lambda +1)}\right)y_2\right],\hfill \end{array}$$

where

$$\psi (\eta ,\tau )=\frac{1-\eta }{2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2}.$$

We obtain following inequality with respect to Lemma 1 and equality given by (2)

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2(2\lambda +1)},0\leqq \left|\psi (\eta ,\tau )\right|\leqq \frac{1}{4(2\lambda +1)},\hfill \\ \\ 2\left|\psi (\eta ,\tau )\right|,\left|\psi (\eta ,\tau )\right|\geqq \frac{1}{4(2\lambda +1)}.\hfill \end{array}\right.$$

After some computations, we obtain

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2(2\lambda +1)};\left|\eta -1\right|\leqq \frac{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}{4(2\lambda +1)},\hfill \\ \\ \frac{2\left|\eta -1\right|}{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|};\left|\eta -1\right|\geqq \frac{\left|2\left(\lambda +1-\lambda \delta (\lambda -1)\right)+(1-\tau ){\left(\lambda +1\right)}^2\right|}{4(2\lambda +1)}.\hfill \end{array}\right.$$

□

For $\delta =\lambda =0$, Theorem 2 gives the results of Cotîrlǎ and Wanas (see [34]).

Corollary 3 

([34]). For $\eta \in \mathbb{R}$, let $f\in {\mathcal{S}}_E^{\ast }\left(\vartheta \right)$ be of the style (1). Then,

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{2};\left|\eta -1\right|\leqq \frac{\left|3-\tau \right|}{4},\hfill \\ \\ \frac{2\left|\eta -1\right|}{\left|3-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|3-\tau \right|}{4}.\hfill \end{array}\right.$$

When $\delta =0$ and $\lambda =1$ Theorem 2 leads to the known result on Cotîrlǎ and Wanas for the family ${\mathcal{C}}_E\left(\vartheta \right)$ (see [34]).

Corollary 4 

([34]). For $\eta \in \mathbb{R}$, let $f\in {\mathcal{C}}_E\left(\vartheta \right)$ be of the style (1). Then,

$$\left|a_3-\eta a_2^2\right|\leqq \left\{\begin{array}{c}\frac{1}{6};\left|\eta -1\right|\leqq \frac{\left|2-\tau \right|}{3},\hfill \\ \\ \frac{\left|\eta -1\right|}{2\left|2-\tau \right|};\left|\eta -1\right|\geqq \frac{\left|2-\tau \right|}{3}.\hfill \end{array}\right.$$

If we take $\eta =1$ in Theorem 2, we obtain the next corollary:

Corollary 5. 

If $f\in {\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ be of the form (1), then we find that

$$\left|a_3-a_2^2\right|\leqq \frac{1}{2(2\lambda +1)}.$$

3. Conclusions

Conclusion Geometric Function Theory has important applications in a variety of mathematical branches involving mathematical physics. Therefore, it is the most attractive research field in complex analysis. Since it is also frequently used in electrostatics, aerodynamics and fluid mechanics to address analytical solutions; various holomorphic functions families have been studied by the relevant researchers. The main aim of this study was to constitute a certain family of ${\mathcal{W}}_{\mathsf{\Sigma }}(\delta ,\lambda ;\vartheta )$ of holomorphic and bi-univalent functions identified by generalized telephone numbers. The Taylor Maclaurin coefficient inequalities for these family functions were found in the current study and the renowned Fekete–Szegö problem was also analyzed. The open inquiry is to determine upper bounds for the general coefficients $|a_n|$, $n\geqq 4$ of the this new family of functions.