Bi-univalent functions of complex order defined by Hohlov operator associated with legendrae polynomial

1.   Introduction, definitions and preliminaries

Let $\mathfrak{A}$ denote the class of functions of the form:

which are analytic in the open unit disk

Further, by $\mathfrak{S}$ we shall denote the class of all functions in $\mathfrak{A}$ which are univalent in $\mathbb{U}.$ Some of the important and well-investigated subclasses of the univalent function class $\mathfrak{S}$ include (for example) the class $\mathfrak{S}^*(\alpha)$ of starlike functions of order $\alpha$ in $\mathbb{U}$ and the class $\mathfrak{K}(\alpha)$ of convex functions of order $\alpha$ in $\mathbb{U}.$ It is well known that every function $f\in \mathfrak{S}$ has an inverse $f^{-1},$ defined by

and

where

A function $f \in \mathfrak{A}$ is said to be bi-univalent in $\mathbb{U}$ if both $f(z)$ and $f^{-1}(z)$ are univalent in $\mathbb{U}.$ Let $\Sigma$ denote the class of bi-univalent functions in $\mathbb{U}$ given by (1.1). Note that the functions

with their corresponding inverses

are elements of $\Sigma.$ This subject has been discussed extensively in the pioneering work by Srivastava et al. ^[31] who revived the study of analytic and bi-univalent functions in recent years. It was followed by many sequels to Srivastava et al. ^[31] (see for example, ^{[3,4,5,20,26,50]}).

An analytic function $f$ is subordinate to an analytic function $g,$ written $f(z) \prec g(z),$ provided there is an analytic function $w$ defined on $\mathbb{U}$ with $w(0) = 0$ and $|w(z)| < 1$ sustaining $f(z) = g(w(z)).$ Lately Ma and Minda ^[23] amalgamated various subclasses of starlike and convex functions for which either of the quantity $\frac{z\ f'(z)}{f(z)}$ or $1 + \frac{z\ f''(z)}{f'(z)}$ is subordinate to a more general superordinate function. For this persistence, they considered an analytic function $\phi$ with positive real part in the unit disk $\mathbb{U}, \phi(0) = 1, \phi'(0) > 0,$ and $\phi$ maps $\mathbb{U}$ onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class of Ma-Minda starlike functions consists of functions $f \in \mathfrak{A}$ satisfying the subordination $\frac{z\ f'(z)}{f(z)} \prec\phi(z).$ Similarly, the class of Ma-Minda convex functions of functions $f \in \mathfrak{A}$ satisfying the subordination $1 + \frac{z\ f''(z)}{f'(z)} \prec \phi(z).$

The convolution or Hadamard product of two functions $f, h\in \mathfrak{A}$ is denoted by $f\ast h$ and is defined as

where $f(z)$ is given by (1.1) and $h(z) = z+\sum\limits_{n = 2}^{\infty }b_{n}z^{n}.$ In terms of the Hadamard product (or convolution), the Dziok-Srivastava linear operator involving the generalized hypergeometric function, was introduced and studied systematically by Dziok and Srivastava ^[9,10] and (subsequently) by many other authors. In our present investigation, we recall a familiar convolution operator $\mathfrak{I}_{a, b, c}$ due to Hohlov ^[16,17], which certainly a very specialized case of the widely- (and extensively-) investigated Dziok-Srivastava operator and also much more general convolution operator, known as the Srivastava-Wright operator ^[32] (also see ^[33]).

For the complex parameters $a, \; b\; {\rm{and}}\; c$ with $c\neq 0, -1, -2, -3, \cdots$, the Gaussian hypergeometric function $_{2}F_{1}(a, b, c; z)$ is defined as

where $(\alpha)_{n}$ is the Pochhammer symbol (or the shifted factorial) defined as follows:

For the positive real values $a, \; b\; {\rm{and}}\; c$ with $c\neq 0, -1, -2, -3, \cdots$, by using the Gaussian hypergeometric function given by (1.4), Hohlov ^[16,17] introduced the familiar convolution operator $\mathfrak{I}_{a, b, c}$ as follows:

where

Hohlov ^[16,17] discussed some interesting geometrical properties exhibited by the operator $\mathfrak{I}_{a, b; c}$. The three-parameter family of operators $\mathfrak{I}_{a, b; c}$ contains, as its special cases, most of the known linear integral or differential operators. In particular, if $b = 1$ in (1.6), then $\mathfrak{I}_{a, b; c}$ reduces to the Carlson-Shaffer operator.Similarly, it is easily seen that the Hohlov operator $\mathfrak{I}_{a, b; c}$ is also a generalization of the Ruscheweyh derivative operator as well as the Bernardi-Libera-Livingston operator.

Recently there has been triggering interest to study bi-univalent function class $\Sigma$ and obtained non-sharp coefficient estimates on the first two coefficients $|a_2|$ and $|a_3|$ of (1.1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients:

is still an open problem(see ^{[3,4,5,20,26,50]}). Many researchers (see ^{[13,15,22,31,51,52]}) have recently introduced and investigated several interesting subclasses of the bi-univalent function class $\Sigma$ and they have found non-sharp estimates on the first two Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|.$

In 1782, Adrien-Marie Legendre discovered Legendre polynomials, which have plentiful physical applications. The Legendre polynomials $P_n(x),$ intermittently called Legendre functions of the first kind, are the particular solutions to the Legendre differential equation

Here and in the following, let $\mathbb{C}$ and $\mathbb{N}$ denote the sets of complex numbers and positive integers, respectively, and let $\mathbb{N}_0 = \mathbb{N}\cup\{0\}.$ The Legendre polynomials are defined by Rodrigues' formula

for arbitrary real or complex values of the variable x. The Legendre polynomials Pn(x) are generated by the following function

where the particular branch of $(1-2xt+t^2)^{-\frac{1}{2}}$ is taken to be 1 as $t\rightarrow 0.$ The first few Legendre polynomials are

A general case of the Legendre polynomials and their applications can be found in ^[18,24]. The function

is in $\mathfrak{P}$ for every real $\alpha$ (see [ ^[14], Page 102], ^[28]), where $\mathfrak{P}$ is the Caratheodory class defined by

$p(z) = 1+c_1z+c_2z^2+\cdots.$ By using (1.8), it is easy to check that

where

In particular by using (1.9), we get

If we consider

From the geometric properties of the Koebe function, the function $\phi$ maps the unit disc onto the right plane $\Re(w) > 0$ minus the slit along the positive real axis from $\frac{1}{|\cos\frac{\alpha}{2}|}$ to $\infty$. $\phi(\mathbb{U})$ is univalent, symmetric with respect to the real axis and starlike with respect to $\phi(0) = 1.$

Motivated by aforementioned study on bi-univalent functions ^{[2,11,13,15,22,29,30,31,51,52]} and present investigation of bi univalent functions associated with various polynomials as well as by many recent works on the Fekete-Szegö functional and other coefficient estimates (see ^{[1,7,25,34,35,36,37,38,39,40,41,42,43,47,48,49]}) in the present paper we introduce new subclasses of the function class $\Sigma$ of complex order $\vartheta \in \mathbb{C}\backslash \{0\}$, involving Hohlov operator $\mathfrak{I}_{a, b; c}$ related with legendrae polynomial and find estimates on the coefficients $|a_2|$ and $|a_3|$ for functions in the new subclasses of function class $\Sigma.$ Several related classes are also considered, and connection to earlier known results are stated.

Definition 1. A function $f\in \Sigma$ given by (1.1) is said to be in the class $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$ if the following conditions are satisfied:

and

where the function $g$ is given by(1.2).

On specializing the parameters $\lambda$ and $a, b, c$ one can state the various new subclasses of $\Sigma$ as illustrated in the following examples.

Example 2. For $\lambda = 1$ and $\vartheta \in \mathbb{C}\backslash \{0\},$ a function $f\in \Sigma,$ given by (1.1) is said to be in the class $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \phi)$ if the following conditions are satisfied:

and

where $z, w \in \mathbb{U}$ and the function $g$ is given by(1.2).

Example 3. For $\lambda = 0$ and $\vartheta \in \mathbb{C}\backslash \{0\},$ a function $f\in \Sigma,$ given by (1.1) is said to be in the class $\mathfrak{G}^{a, b; c}_{\Sigma}(\vartheta, \phi)$ if the following conditions are satisfied:

and

where $z, w \in \mathbb{U}$ and the function $g$ is given by (1.2).

It is of interest to note that for $a = c$ and $b = 1,$ the class $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$ reduces to the following new subclasses

Example 4. For $\lambda = 1$ and $\vartheta \in \mathbb{C}\backslash \{0\},$ a function $f\in \Sigma,$ given by (1.1) is said to be in the class $\mathfrak{S}^*_{\Sigma}(\vartheta, \phi)$ if the following conditions are satisfied:

where $z, w \in \mathbb{U}$ and the function $g$ is given by (1.2).

Example 5. For $\lambda = 0$ and $\vartheta \in \mathbb{C}\backslash \{0\},$ a function $f\in \Sigma,$ given by (1.1) is said to be in the class $\mathfrak{H}^*_{\Sigma}(\vartheta, \phi)$ if the following conditions are satisfied:

where $z, w \in \mathbb{U}$ and the function $g$ is given by (1.2)

In the following section we find estimates on the coefficients $|a_2|$ and $|a_3|$ for functions in the above-defined subclasses $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$ of the function class $\Sigma$ by employing the techniques used earlier by Deniz in ^[11].

In order to derive our main results, we shall need the following lemma.

Lemma 6. (see ^[28]) If $h\in \mathfrak{P},$ then $|c_k|\leqq 2$ for each $k,$ where $\mathfrak{P}$ is the family of all functions $h,$ analytic in $\mathbb{U},$ for which

where

2.   Coefficient bounds for the function class $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$

We begin by finding the estimates on the coefficients $|a_2|$ and $|a_3|$ for functions in the class $\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi).$ Define the functions $p(z)$ and $q(z)$ by

and

or, equivalently,

and

Then $p(z)$ and $q(z)$ are analytic in $\mathbb{U}$ with $p(0) = 1 = q(0).$ Since $u, v : \mathbb{U} \rightarrow \mathbb{U},$ the functions $p(z)$ and $q(z)$ have a positive real part in $\mathbb{U},$ and $|p_{i}| \leq 2$ and $|q_{i}| \leq 2.$

Theorem 7. Let $f$ be given by $(1.1)$ and in the class$\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi).$ Then

and

Proof. It follows from (1.12) and (1.13) that

and

where $p(z)$ and $q(w)$ in $\mathfrak{P}$ and have the following forms:

and

respectively. Now, equating the coefficients in (2.3) and (2.4), we get

and

From (2.7) and (2.9), we find that

which implies

and

Adding (2.8)and (2.10), by using(2.11) and (2.12), we obtain

Thus, by using (1.11)

Applying Lemma 6 for the coefficients $p_2$ and $q_2,$ we immediately have

Hence,

This gives the bound on $|a_2|$ as asserted in (2.1).

Next, in order to find the bound on $|a_3|$, by subtracting (2.10) from (2.8), we get

It follows from (2.11), (2.12) and (2.17) that

Applying Lemma 6 once again for the coefficients $p_2,$ $q_2$ and using (1.11), we readily get

This completes the proof of Theorem 7.

Fixing $\lambda = 1$ in Theorem 7, we have the following corollary.

Corollary 8. Let $f$ be given by $(1.1)$ and in the class$\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \phi).$ Then

and

Taking $a = c$ and $b = 1,$ in Corollary 8, we get the following corollary.

Corollary 9. Let $f$ be given by $(1.1)$ and in the class$\mathfrak{S}^{*}_{\Sigma}(\vartheta, \phi).$Then

and

Fixing $\lambda = 0$ in Theorem 7, we have the following corollary.

Corollary 10. Let $f$ be given by $(1.1)$ and in the class$\mathfrak{G}^{a, b; c}_{\Sigma}(\vartheta, \phi).$Then

and

Taking $a = c$ and $b = 1,$ in Corollary 10, we get the following corollary.

Corollary 11. Let $f$ be assumed by $(1.1)$ and in the class$\mathfrak{H}^{*}_{\Sigma}(\vartheta, \phi).$ Then

and

Due to Zaprawa ^[53], we prove Fekete-Szegö inequalities ^[12] for functions $f\in\mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi).$

Theorem 12. For $\nu\in\mathbb{R},$ let $f$ be given by $(1.1)$ and $f\in \mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$, then

where

Proof. From (2.18), we have

By substituting (2.15) in (2.28), we have

where

Thus by taking modulus of (2.29), we conclude that

where $h(\nu)$ is given by (2.27).

By taking $\nu = 1$ in above Theorem one can easily state the following:

Remark 13. Let the function $f$ be assumed by $(1.1)$ and $f\in \mathfrak{S}^{a, b; c}_{\Sigma}(\vartheta, \lambda, \phi)$. Then

3.   Subclass of bi-univalent function $\mathfrak{M}^{a, b, c}_{\Sigma}(\tau, \phi)$

In ^[27], Obradovic et.al gave some criteria for univalence expressing by $\Re(f'(z)) > 0,$ for the linear combinations

Based on the above definitions recently, Lashin in ^[19] introduced and studied the new subclasses of bi-univalent function.

Definition 14. A function $f\in\Sigma$ given by $(1.1)$ is said to be in the class $\mathfrak{M}^{a, b, c}_{\Sigma}(\tau, \phi)$ if it satisfies the following conditions :

and

where $\tau \geq 1, z, w \in \mathbb{U}$ and the function $g$ is given by (1.2).

Remark 15. For a function $f\in\Sigma$ given by $(1.1)$, is said to be in the class $\mathfrak{M}^{a, b, c}_{\Sigma}(1, \phi)\equiv \mathfrak{K}^{a, b, c}_{\Sigma}(\phi)$ if it satisfies the following conditions :

where $z, w \in \mathbb{U}$ and the function $g$ is given by (1.2).

Theorem 16. Let $f$ be given by $(1.1)$and $f\in\mathfrak{M}^{a, b, c}_{\Sigma}(\tau, \phi)$, $\tau \geq 1$. Then

and

Proof. It follows from (3.1) and (3.2) that

and

From (3.5) and (3.6), we have

and

Now, equating the coefficients, we get

and

From (3.7) and (3.9), we get

From (3.7) by using (1.11),

Also

Thus by (1.11), we get

Now from (3.8), (3.10) and using (3.14), we obtain

Thus, by (3.16) we obtain

From (3.8) from (3.10) and using(3.11), we get

Then taking modulus, we get

Using (3.12) and (3.15), we get

Now by using (3.16) in (3.18),

Due to Zaprawa ^[53], we prove Fekete-Szegö inequalities ^[12] for functions $f\in \mathfrak{M}^{a, b, c}_{\Sigma}(\tau, \phi)$

Theorem 17. For $\nu\in\mathbb{R},$ let $f$ be given by $(1.1)$ and $f\in \mathfrak{M}^{a, b, c}_{\Sigma}(\tau, \phi),$ then

where

Proof. From (3.17), we have

By substituting (3.16) in (3.19), we have

where

Thus by taking modulus of (3.20), we get

where $h(\nu)$ is given by (3.21).

By taking $\nu = 1$ in above Theorem one can easily state the following:

Remark 18. Let $f$ be given by $(1.1)$ and $f\in \mathfrak{M}^{a, b, c}_{\Sigma, }(\tau, \phi)$. Then

4.   Concluding remarks

If $a = 1$, $b = 1+\delta$, $c = 2+\delta$ with $\Re(\delta) > -1$, then the operator $\mathfrak{I}_{a, b, c}f$ turns into familiar Bernardi operator ^[6]

$\mathfrak{I}_{1, 1, 2}f$ and $\mathfrak{I}_{1, 2, 3}f$ are the famous Alexander ^[14] and Libera ^[21] operators, respectively. Further if $b = 1$ in (1.6), then $\mathfrak{I}_{a, b; c}$ immediately yields the Carlson-Shaffer operator $L(a, c)(f) : = \mathfrak{I}_{a, 1, c}f$ ^[8]. So, numerous other interesting corollaries and consequences of our main results (which are asserted by Theorem 7-Theorem 17) can be derived similarly. Further by fixing $\alpha = \pi$ one can state the referents new results for the function classes defined in this paper. The facts involved may be left as an exercise for the interested reader. Also, motivating further researches on the subject-matter of this, we have chosen to draw the attention of the interested readers toward a considerably large number of related recent publications(see, for example, ^{[38,44,45,46]}). and developments in the area of mathematical analysis. In conclusion, we choose to reiterate an important observation, which was presented in the recently-published review-cum-expository review article by Srivastava ( ^[38], p. 340), who pointed out the fact that the results for the above-mentioned or new $q-$ analogues can easily(and possibly trivially)be translated into the corresponding results for the so-called $(p; q)-$analogues(with $0 < |q| < p \leq 1$)by applying some obvious parametric and argument variations with the additional parameter $p$ being redundant.

Acknowledgements

The authors would like to thank the referees for their careful reading and helpful comments.

Conflict of interest

The authors declare no conflict of interest in this paper.