Functional inequalities for several classes of q-starlike and q-convex type analytic and multivalent functions using a generalized Bernardi integral operator

1.   Introduction

Let $\mathcal{A}_{p}$ denote the class of analytic and $p$-valent functions $f(z)$ with the next form

in the open unit disk $\Delta = \{z\in\mathbb{C}: \mid z\mid < 1\}$.

For $f\in\mathcal{A}_{p}$, its $q$-derivative or the $q$-difference $\mathcal{D}_{q}f(z)$ is given by

where the $q$-derivative operator $\mathcal{D}_{q}f(z)$ (refer to ^[13] and ^[14]) of the function $f$ is defined by

provided that $f'(0)$ exists, and the $q$-number $[n]_{q}$ is just $[\chi]_{q}$ when $\chi = n\in\mathbb{N}$, here

Note that $\mathcal{D}_{q}f(z)\longrightarrow f'(z)$ when $q\longrightarrow1_{-}$, where $f'$ is the ordinary derivative of the function $f$.

Consider the generalized Bernardi integral operator $\mathcal{J}^{\eta}_{p, q}:\mathcal{A}_{p}\longrightarrow\mathcal{A}_{p}$ with the next form

Then, for $f\in\mathcal{A}_{p}$, we obtain that

where

Here we remark that if $p = 1$, it is exactly $q$-Bernardi integral operator $\mathcal{J}^{\eta}_{q}$ ^[21]. Further, if $p = 1$ and $q\rightarrow1_{-}$, obviously it is the classical Bernardi integral operator $\mathcal{J}_{\eta}$ ^[5]. In fact, Alexander ^[1] and Libera ^[18] integral operators are special versions of $\mathcal{J}_{\eta}$ for $\eta = 0$ and $\eta = 1$, respectively.

For two analytic functions $f$ and $g$, if there exists an analytic function $h$ satisfying $h(0) = 0$ and $\mid h(z)\mid < 1$ for $z\in\Delta$ so that $f(z) = g(h(z))$, then $f$ is subordinate to $g$, i.e., $f\prec g$.

Let $\Lambda$ be the class of all analytic function $\phi$ via the form

It is well known that the $q$-calculus ^[13,14], even the $(p, q)$-calculus ^[6], is a generalization of the ordinary calculus without the limit symbol, and its related theory has been applied into mathematical, physical and engineering fields (see ^[11,15,25]). Since Ismail et al.^[12] firstly utilized the $q$-derivative operator to investigate the $q$-calculus of the class of starlike functions in disk, there had a great deal of work in this respect; for example, refer to Rehman et al. ^[24] for partial sums of generalized $q$-Mittag-Leffler functions, Srivastava et al. ^[31] for Fekete-Szegö inequality for classes of $(p, q)$-starlike and $(p, q)$-convex functions and ^[27] for close-to-convexity of a certain family of $q$-Mittag-Leffler functions, Seoudy and Aouf ^[26] for the coefficient estimates of $q$-starlike and $q$-convex functions and Uçar ^[33] for the coefficient inequality for $q$-starlike functions. Besides, by involving some special functions and operators or increasing the complexity of function classes, many new subclasses of analytic functions associated with $q$-calculus or $(p, q)$-calculus were considered. Here we may refer to ^[9,23], Ahmad et al. ^[2] for convolution properties for a family of analytic functions involving $q$-analogue of Ruscheweyh differential operator, Dweby and Darus ^[8] for subclass of harmonic univalent functions associated with $q$-analogue of Dziok-Srivastava operator, Mahmmod and Sokól ^[20] for new subclass of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Srivastava et al. ^[28] for coefficient inequalities for $q$-starlike functions associated with the Janowski functions, ^[31] for Fekete-Szegö inequality for classes of $(p, q)$-starlike and $(p, q)$-convex functions using the $q$-Bernardi integral operator and ^[32] for some results on the $q$-analogues of the incomplete Fibonacci and Lucas polynomials. For the multivalent functions, Purohit ^[22] ever studied a new class of multivalently analytic functions associated with fractional $q$-calculus operators, while Shi et al. ^[29] investigated the multivalent $q$-starlike functions connected with circular domain. Moreover, Arif et al. ^[4] considered a $q$-analogue of the Ruscheweyh type operator, and Srivastava et al. ^[30] dealt with basic and fractional q-calculus and associated Fekete-Szegö problems for $p$-valently $q$-starlike functions and $p$-valently $q$-convex functions of complex order using certain integral operators, and Khan et al. ^[17] for a new integral operator in $q$-analog for multivalent functions. Stimulated by the previous results, in the paper we intend to introduce and investigate several new subclasses of $q$-starlike and $q$-convex type analytic and multivalent functions involving a generalized Bernardi integral operator, and establish the corresponding Fekete-Szegö type functional inequalities for these function classes. Besides, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are provided.

From now on we introduce some general subclasses of analytic and multivalent functions associated with the $q$-derivative operator and the generalized Bernardi integral operator.

Definition 1.1. Let $f(z)\in\mathcal{A}_{p}$ and $\mu, \lambda\geq0$. If the following subordination

is satisfied for $z\in\Delta$, then we call that $f(z)$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$.

Definition 1.2. Let $f(z)\in\mathcal{A}_{p}$ and $0\leq\lambda\leq1$. If the following subordination

is satisfied for $z\in\Delta$, then we call that $f(z)$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$.

Definition 1.3. Let $f(z)\in\mathcal{A}_{p}$ and $\mu\geq0$. If the following subordination

is satisfied for $z\in\Delta$, then we call that $f(z)$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$.

Remark 1.4. If we put

or

in Definition (1.1–1.3), then the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$) reduces to $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$) or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$). Without the generalized Bernardi integral operator, the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$) is the classical function class $\mathcal{LN}(\mu, \lambda; \phi)$ (res. $\mathcal{LM}(\lambda; \phi)$ and $\mathcal{NS}(\mu; \phi)$) when $p = 1$ and $q\rightarrow1_{-}$.

Let $\Omega$ be the class of functions $\omega(z)$ denoted by

via the inequality $\vert\omega(z)\vert < 1(z\in\Delta)$. Now we recall some necessary Lemmas below.

Lemma 1.5 (^[16]). Let the function $\omega\in\Omega$. Then

Specially, the sharp result holds for the next function

Lemma 1.6 (^[7,10]). Let $\mathcal{P}$ be the class of all analytic functions $h(z)$ of the following form

satisfying $\Re h(z) > 0$ and $h(0) = 1$. Then there exist the sharp coefficient estimates $\mid c_{n}\mid\leq2(n\in\mathbb{N})$. In Particular, the equality holds for all $n$ for the next function

Lemma 1.7 (^[3,19]). Let the function $\omega\in\Omega$. Then

For $\kappa < -1$ or $\kappa > 1$, the inequality holds literally if and only if $\omega(z) = z$ or one of its rotations. If $< \kappa < 1$, the inequality holds literally if and only if $\omega(z) = z^{2}$ or one of its rotations. In Particular, if $\kappa = -1$, then the sharp result holds for the next function

or one of its rotations. If $\kappa = 1$, then the sharp result holds for the next function

or one of its rotations. If $-1 < \kappa < 1$, then the upper bound is sharp as the followings

and

2.   Functional estimates for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$

By (1.9) we give that

In the section, with Lemma 1.5 we study Fekete-Szegö functional problem for the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$ and provide the following theorem.

Theorem 2.1. Let $\delta\in\mathbb{C}$. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$, then

where

Moreover, the sharp result holds for the next function

Proof. Assume that $f(z)\in\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$. Then, from Definition 1.1, there exists an analytic function $\omega(z)\in\Omega$ such that

Part case

Since

by (2.1) and (2.2) we see that

Thereby

and

Further, with (2.3) and (2.4) we obtain that

where

Therefore, according to Lemma 1.5 we finish the proof of Theorem 2.1.

Corollary 2.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$, then

Moreover, the sharp result holds for the next function

When $\phi\in\mathcal{P}$, combining (2.3) and (2.4) with Lemma 1.6 we instantly establish the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$.

Corollary 2.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$, then

and

If we choose real $\delta$ and $\eta$, then by Lemma 1.7 we derive the next result for Fekete-Szegö problem.

Theorem 2.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$, then

where

and

Moreover, we take

Then, each of the following results is true:

(A) For $\delta\in[\Upsilon_{1}, \Upsilon_{3}]$,

(B) For $\delta\in[\Upsilon_{3}, \Upsilon_{2}]$,

where

Remark 2.5. Fixing the parameter $p = 1$ in Theorems 2.1 and 2.4, we can state the new results for the univalent function classes $\mathcal{LN}^{\eta}_{1, q}(\mu, \lambda; \phi) = \mathcal{LN}^{\eta}_{q}(\mu, \lambda; \phi)$. As Remark 1.4, we may consider $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \alpha)$ or $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \beta)$ to establish latest results. On the other hand, for the different parameters $\mu$ and $\lambda$, we can deduce new results for $\mathcal{LN}^{\eta}_{p, q}(\mu, \lambda; \phi)$.

3.   Functional estimates for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$

In the section we mainly consider Fekete-Szegö functional problem for the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$ and establish the theorem as follows.

Theorem 3.1. Let $\delta\in\mathbb{C}$. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$, then

where

Moreover, the sharp result holds for the next function

Proof. If $f\in\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$, from Definition 1.2 there exists an analytic function $\omega(z)\in\Omega$ such that

Part case

Since

by (2.1) and (3.1) we note that

and

Then, it leads to

and

Furthermore, in accordance with (3.2) and (3.3) we gain that

where

Thus, from Lemma 1.5 we give the Fekete-Szegö functional inequality in Theorem 3.1.

Corollary 3.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$, then

Moreover, the sharp result holds for the next function

If $\phi\in\mathcal{P}$, by (3.2) and (3.3) we take Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$.

Corollary 3.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$, then

and

On the other hand, if we take real $\delta$ and $\eta$, then by Lemma 1.7 we give the next result for Fekete-Szegö problem.

Theorem 3.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$, then

where

and

Moreover, we choose

Then, each of the following results is true:

(A) For $\delta\in[\Gamma_{1}, \Gamma_{3}]$,

(B) For $\delta\in[\Gamma_{3}, \Gamma_{2}]$,

where

Remark 3.5. Similarly, by taking the parameter $p = 1$ in Theorems 3.1 and 3.4, we can obtain the new results for the univalent function classes $\mathcal{LM}^{\eta}_{1, q}(\lambda; \phi) = \mathcal{LM}^{\eta}_{q}(\lambda; \phi)$. As Remark 1.4, we may consider $\mathcal{LM}^{\eta}_{p, q}(\lambda; \alpha)$ or $\mathcal{LM}^{\eta}_{p, q}(\lambda; \beta)$ to establish latest results. Clearly, for special parameter $\lambda$, we can still imply new results for $\mathcal{LM}^{\eta}_{p, q}(\lambda; \phi)$.

4.   Functional estimates for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$

In the section we investigate Fekete-Szegö functional problem for the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$ and obtain the corresponding theorem below.

Theorem 4.1. Let $\delta\in\mathbb{C}$. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$, then

where

Moreover, the sharp result holds for the next function

Proof. Since $f\in\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$, from Definition 1.3 there exists an analytic function $\omega(z)\in\Omega$ such that

Part case

Since

from (2.1) and (4.1) we know that

and

Thus it deduces that

and

Moreover, in the light of (4.2) and (4.3) we know that

where

Hence, in view of Lemma 1.5 we get the Fekete-Szegö functional inequality in Theorem 4.1.

Corollary 4.2. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$, then

Moreover, the sharp result holds for the next function

Once $\phi\in\mathcal{P}$, together with (4.2) and (4.3) we apply Lemma 1.6 to prove the next corollary for the coefficient bounds of $a_{p+1}$ and $a_{p+2}$.

Corollary 4.3. If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$, then

and

Clearly, if we let $\delta$ and $\eta$ be real, then from Lemma 1.7 we also show the following result for Fekete-Szegö problem.

Theorem 4.4. Let $\delta, \eta\in\mathbb{R}$ and $\phi\in\Lambda$ satisfying

If $f(z)\in\mathcal{A}_{p}$ belongs to the class $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$, then

where

and

Moreover, we put

Then, each of the following results is true:

(A) For $\delta\in[\Pi_{1}, \Pi_{3}]$,

(B) For $\delta\in[\Pi_{3}, \Pi_{2}]$,

where

Remark 4.5. Similarly, by choose the parameter $p = 1$ in Theorems 4.1 and 4.4, we can provide the new results for the univalent function classes $\mathcal{NS}^{\eta}_{1, q}(\mu; \phi) = \mathcal{NS}^{\eta}_{q}(\mu; \phi)$. As Remark 1.4, we may consider $\mathcal{NS}^{\eta}_{p, q}(\mu; \alpha)$ or $\mathcal{NS}^{\eta}_{p, q}(\mu; \beta)$ to establish latest results. Besides, for the fixed parameter $\mu$, we can still infer new results for $\mathcal{NS}^{\eta}_{p, q}(\mu; \phi)$.

5.   Conclusion

By involving a generalized Bernardi integral operator, several new subclasses of $q$-starlike and $q$-convex type analytic and multivalent functions are introduced to generalize the classical starlike and convex functions. Meanwhile, for these classes we may know integral operator and $q$-derivative as well as multivalency how to change the coefficients of functions. In our main results, we establish the Fekete-Szegö type functional inequalities for these function classes. Further, the corresponding bound estimates of the coefficients $a_{p+1}$ and $a_{p+2}$ are interpreted. In fact, if we use the other integral operators or take $(p, q)$-operator when certain function is univalent but not multivalent, we may get many similar results as in this article.

Acknowledgments

We thank the referees for their careful readings and using comments so that this manuscript is greatly improved. This work is supported by Institution of Higher Education Scientific Research Project in Ningxia of the People's Republic of China under Grant NGY2017011, Natural Science Foundation of Ningxia of the People's Republic of China under Grant 2020AAC03066, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grant 2018MS01026, the Natural Science Foundation of the People's Republic of China under Grant 11561001, 42064004 and 11762016.

Conflict of interest

The authors declare no conflict of interest.