A study of the Fekete-Szegö functional and coefficient estimates for subclasses of analytic functions satisfying a certain subordination condition and associated with the Gegenbauer polynomials

1.   Introduction, definitions and motivation

Let $\mathcal{A}$ denote the family of all analytic functions, which are defined on the open unit disk

and normalized by the following condition:

Such functions $f \in \mathcal{A}$ have the Taylor-Maclaurin series expansion given by

Furthermore, by ${\mathcal{S}}$ we denote the class of all functions $f\in {\mathcal{A}}$ that are also univalent in ${\mathbb{U}}.$

With a view to recalling the principle of subordination between analytic functions, let the functions $f\left(z\right)$ and $g\left(z\right)$ be analytic in ${\mathbb{U}}.$ We then say that the function $f\left(z\right)$ is subordinate to $g\left(z\right)$ in ${\mathbb{U}},$ if there exists a Schwarz function $w\left(z\right),$ analytic in ${\mathbb{U}}$ with

such that

We denote this subordination by

If the function $g$ is univalent function in ${\mathbb{U}},$ then

The concept of arithmetic means of functions and other entities is frequently used in mathematics, especially in geometric function theory of complex analysis. Making use of the concept of arithmetic means, Mocanu ^[12] introduced the class of $\alpha$-convex functions $\left(0\leqq \alpha \leqq 1\right)$ as follows:

which, in some case, corresponds to the class of starlike functions and, in another case, to the class of convex functions. In general, the class of $\alpha$-convex functions determines the arithmetic bridge between starlikeness and convexity.

By using the geometric means, Lewandowski et al. ^[9] defined the class of $\mu$-starlike functions $\left(0\leqq \mu \leqq 1\right)$ consisting of functions $f\in {\mathcal{A}}$ that satisfy the following inequality:

We note that the class of $\mu$-starlike functions constitutes the geometric bridge between starlikeness and convexity.

We now recall that a function $f\in {\mathcal{A}}$ maps ${\mathbb{U}}$ onto a starlike domain with respect to $w_{0} = 0$ if and only if

On the other hand, a function $f\in {\mathcal{A}}$ maps $\mathbb{U}$ onto a convex domain if and only if

It is well known that, if a function $f\in {\mathcal{A}}$ satisfies $\left(1.4\right)$, then $f$ is univalent and starlike in ${\mathbb{U}}$.

Let $\beta \in \left[0, 1\right).$ A function $f\in {\mathcal{A}}$ is said to be starlike of order $\beta$ and convex of order $\beta,$ if

and

respectively.

In the year 1933, Fekete and Szegö ^[6] obtained a sharp bound of the functional $a_{3}-\nu a_{2}^{2},$ with real $\nu$ $\left(0\leqq \nu \leqq 1\right)$ for a univalent function $f.$ Since then, the problem of finding the sharp bounds for the Fekete-Szegö functional of any compact family of functions or for $f\in {\mathcal{A}}$ with any complex $\nu$ is known as the classical Fekete-Szegö problem (see, for details, ^[14,21]). More recently, in the year 1994. Szynal ^[25] introduced and investigated the class $\mathcal T\left(\lambda \right)\; \; (\lambda \geqq 0)$ as a subclass of ${\mathcal{A}}$ consisting of functions of the form

where

and $\sigma$ is a probability measure on the interval $\left[-1, 1\right].$ The collection of such measures on $\left[a, b\right]$ is denoted by $P_{\left[a, b\right]}.$ The function $k\left(z, t\right)$ has the following Taylor-Maclaurin series expansion:

where $C_{n}^{\left(\lambda \right)}\left(t\right)$ denotes the Gegenbauer (or ultraspherical) polynomials of order $\lambda$ and degree $n$ in $t$, which are generated by (see, for details, ^[18])

If a function $f\in \mathcal T\left(\lambda \right)$ is given by $\left(1.8\right),$ then the coefficients of this function can be written as follows:

We note that $\mathcal T\left(1\right) = :\mathcal T$ is the well-known class of typically real functions.

The Gengenbauer (or ultraspherical) polynomials $C_n^{(\lambda)}(t)$ as well as their relatively more familiar special or limit cases such as the Legendre (or spherical) polynomials $P_n(t)$, the Chebyshev polynomials $T_n(t)$ of the first kind, and the Chebyshev polynomials $U_n(t)$ of the second kind, are orthogonal over the interval $[-1, 1]$. In fact, we have

The subject of Geometric Function Theory of Complex Analysis has been a fast-growing area of research in recent years. Noteworthy developments and studies involving various old (or traditional) as well as newly-introduced subclasses of the class of normalized analytic or meromorphic functions, together with the multivalent analogues in each case, can be found in the remarkably vast literature on this subject. A good source for some recent researches and developments in Geometric Function Theory of Complex Analysis is the 888-page edited volume by Milovanović and Rassias ^[11].

Our present investigation is motivated by the above-mentioned developments as well as by many recent works on the Fekete-Szegö functional and other coefficient estimate problems by (for example) Dziok et al. ^[5], Ałtinkaya and Yalçin ^[2], Srivastava et al. ^[20], Szatmari and Ałtinkaya ^[24], and Çağlar et al. ^[4] (see also ^{[1,3,8,10,13,14,17,19,21]}). Here, in this paper, we introduce and study a new subclass of normalized analytic functions $\mathcal{A}$ in $\mathbb{U}$, which we denote by

We say that a function $f\in\mathcal{A}$ of the form (1.1) is in the following class$:$

if it satisfies the following subordination condition associated the Gegenbauer (or ultraspherical) polynomials $C_{n}^{\left(\lambda\right)}(t)$ of order $\lambda$ and degree $n$ in $t$:

where $H\big(z, C_{n}^{\left(\lambda\right)}\left(t\right)\big)$ is given by the generating relation (1.11),

and

For functions in this subclass, we first derive the estimates for the initial Taylor-Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ and then examine the corresponding Fekete-Szegö inequality. Finally, the results obtained are applied to subclasses of normalized analytic functions satisfying the subordination condition and associated with the Legendre and Chebyshev polynomials. In the concluding section, we have indicated the possibility of using the basic or quantum (or $q$-) calculus and we have also exposed the so-called trivial and inconsequential $(p, q)$-variations by forcing-in an obviously redundant (or superfluous) parameter $p$ in the familiar $q$-calculus.

2.   Initial coefficient bounds for the function class $\mathcal F_{\left(\beta, \gamma\right)} \bigg(\alpha, \delta, \mu, H\big(z, C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg)$

Our first result (Theorem 1 below) provides bounds for the initial Taylor-Maclaurin coefficients $a_2$ and $a_3$ in (1.1).

Theorem 1. Let the function $f(z)$ given by $(1.1)$ be in the following class$:$

Then

and

provided that

Proof. Under the hypotheses of Theorem 1, we find from (1.1) and (1.15) that

Now, upon setting $\nabla: = 2\gamma \beta +\gamma -\beta$ in (2.3), we can write

which readily yields

and

If we make use of the above expressions and apply (1.14), we see that

for some analytic function $p(z)$ given by

such that

Then, for all $j\in {\mathbb N}$, we have

Also, for all $\xi \in {\mathbb R}$, we obtain

It follows from (2.4) that

which leads us to the following consequences:

and

Now, from $\left(1.11\right),$ $\left(2.5\right)$ and $\left(2.8\right)$, we can write

We thus obtain the first coefficient bound (2.1) asserted by Theorem 1:

Similarly, from $\left(1.11\right)$, $\left(2.5\right)$ and $\left(2.9\right)$, we can show that

which, in conjunction with $\left(2.6\right)$, yields

Finally, by making use of the parametric constraints given with Theorem 1, we find eventually that

which is precisely the coefficient bound (2.2) of Theorem 1. This completes our proof of Theorem 1.

The following corollaries and consequences of Theorem 1 are worthy of note.

Ⅰ. If we set $\alpha = \delta = 1$ or $\alpha = \mu-1 = 0$ in Theorem 1, we obtain the following corollary.

Corollary 1. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then

and

provided that

Ⅱ. Taking $\beta = \gamma = 0$ in Theorem 1, we obtain the following corollary.

Corollary 2. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then

and

provided that

Ⅲ. If we put $\delta-1 = \mu = 0$ in Theorem 2.1, we obtain the following corollary.

Corollary 3. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then

and

provided that

Ⅳ. Taking $\delta-1 = \mu = 0$ and $\beta = \gamma = 0$ in Theorem 1, we obtain the following corollary.

Corollary 4. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then

and

provided that

Ⅴ. If we set $\alpha = \beta = \gamma = 0$ in Theorem 2.1, we obtain the following corollary.

Corollary 5. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then

and

provided that

3.   Fekete-Szegö inequality for the function class $\mathcal F_{\left(\beta, \gamma\right)} \bigg(\alpha, \delta, \mu, H\big(z, C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg)$

In this section, we find the sharp bounds of the Fekete-Szegö functional $a_{3}-\xi a_{2}^{2}$ defined for functions $f\in \mathcal F_{\left(\beta, \gamma\right)} \bigg(\alpha, \delta, \mu, H\big(z, C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg),$ which are given by $\left(1.1\right)$.

Theorem 2. Let the function $f\left(z\right)$ given by $\left(1.1\right)$ be in the following class$:$

Then$,$ for some $\xi \in {\mathbb R},$

where

and

such that

and

$\delta$ being given by

Proof. If the above expressions for $K$, $B$ and $R$ are used for those in the Eqs (2.1) and (2.2), we get

and

Now, from (3.2) and (3.3), we can easily see that

and

Therefore, in view of (2.6), we conclude that

Finally, by using the generating function (1.11) in (3.4), we get

Moreover, since $t > 0$, we have

which evidently completes the proof of Theorem 2.

Just as we deduced several consequences of Theorem 1 in the preceding section, here we deduce the following analogous corollaries of Theorem 2.

Ⅰ. Taking $\alpha = \delta = 1$ or $\alpha = \mu-1 = 0$ in Theorem 2, we obtain the following corollary.

Corollary 6. Let the function $f\left(z\right)$ given by (1.1) be in the following class$:$

Then$,$ for some $\xi \in {\mathbb R},$

where

and

$\delta$ being given by

Ⅱ. Upon setting $\beta = \gamma = 0$ in Theorem 2, we are led to the following corollary.

Corollary 7. Let the function $f\left(z\right)$ given by (1.1) be in the following class$:$

Then$,$ for some $\xi \in {\mathbb R},$

where

and

such that

and

Ⅲ. Putting $\delta-1 = \mu = 0$ in Theorem 3.1, we get the following corollary.

Corollary 8. Let the function $f\left(z\right)$ given by (1.1) be in the following class$:$

Then, for some $\xi \in {\mathbb R},$

where

and

such that

4.   Applications associated with the Legendre and Chebyshev polynomials

In order to apply our main results in Section 2 and Section 2 to the corresponding function classes associated with the Legendre polynomials $P_n(t)$, the Chebyshev polynomials $T_n(t)$ of the first kind and the Chebyshev polynomials $U_n(t)$ of the second kind, we can make use of their relationships in (1.13) with the Gegenbauer (or ultraspherical) polynomials $C_n^{(\lambda)}(t)$. For example, if we set $\lambda = \frac{1}{2}$, Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8, would readily yield the corresponding results for the function classes associated with the Legendre polynomials $P_n(t)$. In a similar manner, upon setting $\lambda = 1$, we can easily derive the corresponding results for the function classes associated with the Chebyshev polynomials $U_n(t)$ of the second kind. The analogous derivations in respect of the Chebyshev polynomials $T_n(t)$ of the first kind would obviously involve limit processes. Thus, except possibly in the case of the function class associated with the Chebyshev polynomials $T_n(t)$ of the first kind, it is fairly straightforward to set $\lambda = \frac{1}{2}$ and $\lambda = 1$ in Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8, in order to deduce the corresponding assertions for the function classes associated, respectively, with the Legendre polynomials $P_n(t)$ and the Chebyshev polynomials $U_n(t)$ of the second kind. We, therefore, choose to leave all such applications of Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8, as an exercise for the interested reader.

Some of the known special cases of Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8, are being listed below.

Ⅰ. The special case of Theorem 1 when $\lambda = 1$ was given in ^[7].

Ⅱ. If we further put $\lambda = 1$ in Corollary 2, we can derive a known result (see ^[24]).

Ⅲ. Upon setting $\lambda = 1$ in Corollary 5, we are led to a known result (see ^[2]).

Ⅳ. In the special case of Theorem 1 when $\lambda = \alpha = \delta = 1$, we get a known result (see ^[4]).

Ⅴ. The special case of Theorem 2 when $\lambda = 1$ yields a known result (see ^[7]).

Ⅵ. For $\lambda = 1$, Corollary 7 yields a known result (see ^[24]).

Ⅶ. In its special case when $\lambda = 1$, if we further set $\alpha = 0$, we are led to a known result (see ^[2]).

Ⅷ. For $\lambda = \alpha = \delta = 1$, Theorem 2 reduces to a known result (see ^[4]).

Other (known or new) special cases and consequences of our main results asserted by Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8, can be deduced fairly easily. We omit the details involved in these derivations.

5.   Conclusions and observations

Motivated by several interesting developments on the subjects, here we have introduced and investigated the following new subclass of normalized analytic functions in the open unit disk $\mathbb{U}$:

which satisfy a certain subordination condition and are associated with the Gegenbauer (or ultraspherica) polynomials $C_{n}^{\left(\lambda\right)}(t)$ of order $\lambda$ and degree $n$ in $t$. For functions belonging to this function class, we have derived the estimates for the initial Taylor-Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ and we have also examined the Fekete-Szegö functional. Our main results are asserted by Theorem 1 and its Corollaries 1 to 5, as well as Theorem 2 and its Corollaries 6 to 8. It is also shown how some of these main results can be applied to (known or new) subclasses of normalized analytic functions satisfying the corresponding subordination condition and associated with the Legendre polynomials $P_n(t)$, the Chebyshev polynomials $T_n(t)$ of the first kind, and the Chebyshev polynomials $U_n(t)$ of the first kind.

In several recent developments on the Taylor-Maclaurin coefficient estimate problem and the Fekete-Szegö coefficient inequality problem, use has been made successfully of the Horadam polynomials $h_{n}(t)$ which are given by the following recurrence relation$:$

with

for some real constants $a, b, \mathfrak p$ and $\mathfrak q$ (see, for details, ^[16,22,23]; see also the references to the earlier works which are cited in each of these references). Indeed, as its special cases, the Horadam polynomials $h_{n}(t)$ contain a remarkably large number of other relatively more familiar polynomials including (for example) the Fibonacci polynomials, the Lucas polynomials, and the Pell-Lucas polynomials, as well as the Chebyshev polynomials $T_{n}(t)$ of the first kind and the Chebyshev polynomials $U_{n}(t)$ of the first kind. Most (if not all) of these recent developments also apply the basic or quantum (or $q$-) calculus as well. A possible presumably open problem for future researches emerging from our present investigation would involve the analogous usage of the Horadam polynomials $h_n(t)$ instead of the Gegenbauer (or ultraspherical) polynomials $C_n^{(\lambda)}(t)$ which we have used in our investigation.

In concluding this paper, we recall a recently-published survey-cum-expository review article in which Srivastava ^[14] explored the mathematical applications of the $q$-calculus, the fractional $q$-calculus and the fractional $q$-derivative operators in Geometric Function Theory of Complex Analysis, especially in the study of Fekete-Szegö functional. Srivastava ^[14] also exposed the not-yet-widely-understood fact that the so-called $(p, q)$-variation of the classical $q$-calculus is, in fact, a rather trivial and inconsequential variation of the classical $q$-calculus, the additional parameter $p$ being redundant or superfluous (see, for details, [14,p. 340]; see also [15,pp. 1511–1512]).

Conflicts of interest

The authors declare that they have no conflicts of interest.