Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity

1.   Introduction and definitions

Let $\mathcal{A}$ denote the class of all functions $f$ which are analytic in the open unit disk $E = \{z\in \mathbb{C} :\left \vert z\right \vert < 1\}$ and has the Taylor series expansion of the form

Let $\mathcal{S}$ be the subclass of all functions in $\mathcal{A}$ which are univalent in $E$ (see ^[1]). Goodman ^[2] introduced $\mathcal{ UCV}$ of the uniformly convex functions and $\mathcal{ST}$ of starlike functions. A function $f\in \mathcal{A}$ is called uniformly convex if every (positively oriented) circular arc of the form $\left\{ z\in E:\left\vert z-\xi \right\vert = r\right\}$ and $\xi \in E,$ the arc $f(\xi)$ is convex. For more details of the class $\mathcal{UCV}$ and $\mathcal{ST}$ see ^[3].

Later in ^[4] Kanas and Wisniowska introduced the class $k\mathcal{-UCV}$ and the class $k-\mathcal{ST},$ defined as:

and

Note that $f(z)\in k-\mathcal{UCV}\Longleftrightarrow zf^{^{\prime }}(z)\in k-\mathcal{ST}.$

In ^[4], if $k\geq 0,$ the class $k-\mathcal{UCV}$ is defined geometrically as a subclass of univalent functions which map the intersection of $E$ with any disk center et $\zeta$, $\left\vert \zeta \right\vert \leq k,$ onto a convex domain. Therefore, the notion of $k$- uniform convexity is a generalization of the notion of convexity. For $k = 0,$ the center $\zeta$ is the origin and the class $k-\mathcal{UCV}$ reduces to the class $\mathcal{C}$ of convex univalent functions, (see ^[1]). Moreover for $k = 1$ it coincides with the class of uniformly convex functions ($\mathcal{UCV}$) introduced by Goodman ^[2] and studied extensively by Ronning ^[5] and Ma and Minda ^[3]. We note that the class $k- \mathcal{UCV}$ started much earlier in ^[6] with some additional conditions but without the geometric interpretation.

We say that a function $f\in \mathcal{A}$ is in the class $\mathcal{S} _{k, \gamma }^{\ast },$ $k\geq 0,$ $\gamma \in \mathbb{C} \backslash \{0\},$ if and only if

For more detail about the class $\mathcal{S}_{k, \gamma }^{\ast }$, (see ^[7]).

If $f(z)$ and $g(z)$ are analytic in $E$, we say that $f(z)$ is subordinate to $g(z)$, written as $f(z)\prec g(z)$, if there exists a Schwarz function $w(z)$, which is analytic in $E$ with $w(0) = 0$ and $|w(z)| < 1$ such that $f(z) = g(w(z))$. Furthermore, if the function $g(z)$ is univalent in $E$, then we have the following equivalence, (see ^[1])$.$

For two analytic functions

The convolution (Hadamard product) of $f(z)$ and $g(z)$ is defined as:

Let $\mathcal{P}$ denote the well-known Carathéodory class of functions $p$, analytic in the open unit disk $E$, which are normalized by

such that

We have discussed above that Kanas and Wisniowska ^[4] introduced and studied the class $k-\mathcal{UCV}$ of $k$-uniformly convex functions and the corresponding class $k-\mathcal{ST}$ of $k$-starlike functions. Then Kanas and Wisniowska ^[4] defined these classes subject to the conic domain $\Omega _{k},$ ($k\geq 0$) as follows:

or

This domain represents the right half plane for $k = 0$, a hyperbola for $0 < k < 1$, a parabola for $k = 1$ and an ellipse for $k > 1$. Deniz et al. ^[8] defined new subclasses of analytic functions subject to the conic domain $\Omega _{k, },$ (also see ^[9]). Theses classes were then generalized to $\mathcal{KD}(k, \gamma)$ and $\mathcal{SD}(k, \gamma)$ respectively by Shams et al. ^[10] subject to the conic domain $\Omega _{k, \gamma }$ ($k\geq 0$)$,$ $0\leq \gamma < 1$, which is

or

For this conic domains, the following function play the role of extremal function.

where $i\in (0, 1)$, $k = \cosh \left(\frac{\pi K^{^{\prime }}(i)}{4K(i)} \right),$ $K(i)$ is the first kind of Legendre's complete elliptic integral. For details (see ^[4]). Indeed, from (1.2), we have

where

The quantum (or $q$-) calculus is an important tools used to study various families of analytic functions and has inspired the researchers due to its applications in mathematics and some related areas. Srivastava ^[11] studied univalent functions using $q$-calculus. The quantum (or $q$ -)calculus is also widely applied in the approximation theory, especially for various operators, which include convergence of operators to functions in a real and complex domains. Jackson ^[12] was among the first few researchers who defined the $q$-analogue of derivative and integral operator as well as provided some of their applications. Later on, Aral and Gupta ^[13] introduced the $q$-Baskakov-Durrmeyer operator by using $q$-beta function while ^[14] studied the $q$-generalization of complex operators known as $q$-Picard and $q$-Gauss-Weierstrass singular integral operators. Kanas and Raducanu ^[15] introduced the $q$-analogue of Ruscheweyh differential operator and Arif et al. ^[16] discussed some of its applications for multivalent functions while ^[17] studied $q$ -calculus by using the concept of convolution. Authors in ^[18] and ^[19] studied $q$-differential and $q$-integral operators for the class of analytic functions. Here we will present the basic definitions of quantum (or $q$-) calculus which will help us in onwards study.

Definition 1. (^[20]). The $q$-number $[t]_{q}$ for $q\in (0, 1)$ is defined as:

Definition 2. The $q$-factorial $[n]_{q}!$ for $q\in (0, 1)$ is defined as:

Definition 3. The $q$-generalized Pochhammer symbol $[t]_{n, q}$, $t\in \mathbb{C},$ is defined as:

Furthermore, the $q$-Gamma function be defined as:

Definition 4. (^[12]). For $f\in \mathcal{A},$ the $q$-derivative operator or $q$ -difference operator be defined as:

From (1.1) and (1.8), we have

For $n\in \mathbb{N}$ and $z\in E,$ we have

We can observe that

Definition 5. (^[21]). A function $f\in \mathcal{A}$ is said to belong to the class $S_{q}^{\ast }$ if

and

Equivalently, we can rewrite the conditions in (1.9) and (1.10) as follows, (see ^[22]).

Now, making use of quantum (or $q$-) calculus and principle of subordination we present the following definition as:

Definition 6. Let $k\in \lbrack 0, \infty),$ $q\in \left(0, 1\right)$ and $\gamma \in \mathbb{C} \backslash \{0\}.$ A function $p(z)$ is said to be in the class $k-\mathcal{P }_{q, \gamma }$ if and only if

where

and $p_{k, \gamma }(z)$ is given by (1.2).

Geometrically, the function $p(z)\in k-\mathcal{P}_{q, \gamma }$ takes all values from the domain $\Omega _{k, q, \gamma }$ which is defined as follows:

where

The domain $\Omega _{k, q, \gamma }$ represents a generalized conic region.

Remark 1. When $q\rightarrow 1^{-},$ then $\Omega _{k, q, \gamma } = \Omega _{k, \gamma },$ where $\Omega _{k, \gamma }$ is the conic domain considered by Shams et al ^[10].

Remark 2. When $\gamma = 1,$ $q\rightarrow 1^{-},$ then $\Omega _{k, q, \gamma } = \Omega _{k},$ where $\Omega _{k}$ is the conic domain considered by by Kanas and Wisniowska ^[7].

Remark 3. For $\gamma = 1,$ $q\rightarrow 1^{-},$ then $k-\mathcal{P}_{q, \gamma } = \mathcal{P(}p_{k}\mathcal{)},$ where $\mathcal{P(}p_{k}\mathcal{)}$ is the well-known class introduced by Kanas and Wisniowska ^[7].

Remark 4. For $\gamma = 1,$ $k = 0,$ $q\rightarrow 1^{-},$ then $k-\mathcal{P}_{q, \gamma } = \mathcal{P},$ where $\mathcal{P}$ is the well-known class of analytic functions with positive real part.

Definition 7. A function $f\in \mathcal{A}$ is said to be in class $k-\mathcal{UST} (q, \gamma)$ if it satisfies the condition

or equivalently

where

Special cases:

i. For $q\rightarrow 1^{-}$, then the class $k-\mathcal{UST}(q, \gamma)$ reduces to the $\mathcal{S}_{k, \gamma }^{\ast }$ (see ^[7]).

ii. For $\gamma = 1$ and $q\rightarrow 1^{-},$ then the class $k-\mathcal{UST} (q, \gamma)$ reduces to the $k-\mathcal{UCV}$ (see ^[4]).

Geometrically a function $f(z)\in \mathcal{A}$ is said to be in the class $k- \mathcal{UST}(q, \gamma)$, if and only if the function $\mathcal{J}(q, f(z))$ takes all values in the conic domain $\Omega _{k, q, \gamma }$ given by (1.13). Taking this geometrical interpretation into consideration, one can rephrase the above definition as:

Definition 8. A function $f\in \mathcal{A}$ is said to be in the class $k-\mathcal{UST} (q, \gamma)$ if and only if

where $p_{k, \gamma, q}(z)$ is defined by (1.12).

We also set $k-\mathcal{UST}^{-}(q, \gamma) = k-\mathcal{UST}(q, \gamma)\cap T,$ $T$ is the subclass of $k-\mathcal{UST}(q, \gamma)$ consisting of functions of the form

2.   Set of lemmas

In order to prove our main results in this paper, we need each of the following lemmas.

Lemma 1. (see ^[23]). Let $p(z) = 1+\sum \limits_{n = 1}^{\infty }p_{n}z^{n}\prec F(z) = 1+\sum \limits_{n = 1}^{\infty }C_{n}z^{n}.$ If $F(z)$ is convex univalent in $E,$ then

Lemma 2. Let $k\in \lbrack 0, \infty)$ be fixed and

Then

where $Q_{1},$ and $Q_{2}$ is given by $\left(1.4\right)$ and $\left(1.5\right).$

Proof. From (1.12), we have

By using (1.3a) in (2.1), we have

The series $\sum \limits_{n = 1}^{\infty }\frac{2\left(-1\right) ^{n-1}(1-q)^{n-1}}{\left(1+q\right) ^{n}},$ $\sum \limits_{n = 1}^{\infty } \frac{2n\left(-1\right) ^{n-1}(1-q)^{n-1}}{\left(1+q\right) ^{n+1}},$ and $\sum \limits_{n = 1}^{\infty }\frac{2(2n-1)\left(-1\right) ^{n-1}(1-q)^{n}}{ \left(1+q\right) ^{n+1}}$ are convergent and convergent to $1,$ $\frac{2}{ 1+q},$ and $\frac{2(1-q)}{\left(1+q\right) }.$

Therefore (2.2) becomes

This complete the proof of Lemma 2.

Remark 5. When $q\rightarrow 1^{-},$ the Lemma 2, reduces to the lemma which was introduced by Sim et. al ^[24].

Lemma 3. Let $p(z) = 1+\sum \limits_{n = 1}^{\infty }p_{n}z^{n}\in k-\mathcal{P }_{q, \gamma },$ then

Proof. By definition (6), a function $p(z)\in k-\mathcal{P}_{q, \gamma }$ if and only if

where $k\in \lbrack 0, \infty),$ and $p_{k, \gamma }(z)$ is given by (1.2).

By using (2.3) in (2.4), we have

Now by using Lemma 1 on (2.5), we have

Hence the proof of Lemma 3 is complete.

Remark 6. When $q\rightarrow 1^{-},$ then Lemma 3 reduces to the lemma which was introduced by Noor et. al ^[25].

Lemma 4. ^[26]. Let $h(z) = 1+\sum \limits_{n = 1}^{\infty }c_{n}z^{n}$ and $h(z)$ be analytic in $E$ and satisfy $Re\{h(z)\} > 0$ for $z$ in $E$, then the following sharp estimate holds;

3.   Main results

Theorem 1. If a function $f\in \mathcal{A}$ of the form (1.1) and it satisfies

then $f(z)\in k-\mathcal{UST}(q, \gamma).$

Proof. Assume that $\left(3.1\right)$ is holds, then it is suffice to show that

Using (1.16), we have

The last expression is bounded above by 1.

After some simple calculation we have

Hence we complete the proof of Theorem 1.

When $q\rightarrow 1^{-}$ and $\gamma = 1-\alpha$ with $0\leq \alpha < 1$, we have the following known result proved by Shams et. al in ^[10].

Corollary 1. A function $f\in \mathcal{A}$ and of the form (1.1) is in the class $k-\mathcal{UST}(1-\alpha)$ if it satisfies the condition

where $0\leq \alpha < 1$ and $k\geq 0$.

Inequality (3.1) gives us a tool to obtain some special member of $k- \mathcal{UST}(q, \gamma).$ Thus we have the following corollary:

Corollary 2. Let $0\leq k < \infty,$ $q\in \left(0, 1\right)$ and $\gamma \in \mathbb{C} \backslash \{0\}.$ If the inequality

holds for $f(z) = z+a_{n}z^{n},$ then $k-\mathcal{UST}(q, \gamma).$ In particular,

and

Theorem 2. If $f(z)\in k-\mathcal{UST}(q, \gamma)$ and is of the form $\left(1.1\right)$. Then

and

where $Q_{1}$ and $\varphi _{j}$ are defined by $\left(1.4\right)$ and $\left(3.6\right)$.

Proof. Let

Now from (3.4), we have

which implies that

Equating coefficients of $z^{n}$ on both sides, we have

This implies that

By using Lemma 3, we have

where

Now we prove that

For this we use the induction method. For $n = 2$ from (3.5) we have

From (3.3) we have

For $n = 3$, from (3.5), we have

From (3.3), we have

Let the hypothesis be true for $n = m$. From (3.5), we have

From (3.3) we have

By the induction hypothesis, we have

Multiplying $\frac{\left \vert Q_{1}\right \vert +q(q+1)[m-1, q]}{q\left(1+q\right) [m-1]_{q}}$ on both sides of (3.8), we have

That is,

which shows that inequality (3.8) is true for $n = m+1$. Hence the proof of Theorem 2 is complete

When $q\rightarrow 1^{-}$, then we have the following known result, proved by Kanas and Wisniowska in ^[4].

Corollary 3. If $f(z)\in k-\mathcal{UST}(q, \gamma)$ and is of the form $\left(1.1 \right)$. Then

Theorem Let $0\leq k < \infty,$ $q\in (0, 1)$, be fixed and let $f(z)\in k- \mathcal{UST}(q, \gamma)$ and is of the form $\left(1.1\right)$. Then for a complex number $\mu$,

where $v$ is given by $\left(3.13\right)$.

Proof. If $\ f(z)\in k-\mathcal{UST}(q, \gamma),$ then there exist a Schwarz function $w(z)$ with $w(0) = 0$ and $|w(z)| < 1,$ such that

Let $h(z)\in \mathcal{P}$ be a function defined as:

This gives

and

By using (3.11) in (3.10) we obtain

and

For any complex number $\mu$ we have

where

Now by using Lemma $4$ on $\left(3.12\right)$ we have

Hence we complete the proof of Theorem

Theorem 4. Let $k\in \lbrack 0, \infty),$ $q\in \left(0, 1\right)$ and $\gamma \in \mathbb{C} \backslash \{0\}.$ A necessary and sufficient condition for $f(z)$ of the form (1.18) to be in the class $k-\mathcal{UST}^{-}(q, \gamma)$ can be formulated as follows:

The result is sharp for the function

Proof. In view of Theorem 1, it remains to prove the necessity. If $f\in k- \mathcal{UST}^{-}(q, \gamma),$ then in virtue of the fact that $\left \vert \Re (z)\right \vert \leq \left \vert z\right \vert,$ for any $z$, we have

Letting $z\rightarrow 1^{-},$ along the real axis, we obtain the desired inequality (3.14). Hence we complete the proof of Theorem 4.

Corollary 4. Let the function $f(z)$ of the form (1.18) be in the class $k-\mathcal{ UST}^{-}(q, \gamma)$. Then

Corollary 5. Let the function $f(z)$ of the form (1.18) be in the class $k-\mathcal{ UST}^{-}(q, \gamma)$. Then

Theorem 5. Let $k\in \lbrack 0, \infty),$ $q\in \left(0, 1\right)$ and $\gamma \in \mathbb{C} \backslash \{0\}$ and let

and

Then $f\in k-\mathcal{UST}^{-}(q, \gamma),$ if and only if $f$ can be expressed in the form of

Proof. Suppose that

Then

and we find $k-\mathcal{UST}^{-}(q, \gamma).$

Conversely, assume that $k-\mathcal{UST}^{-}(q, \gamma).$ Since

we can set

and

Then

The proof of Theorem 5 is complete.

Theorem 6. Let $k\in \lbrack 0, \infty),$ $q\in (0, 1),$ and $\gamma \in \mathbb{C} \backslash \{0\}.$ Let $\ f$ defined by (1.18) belongs to the class $k- \mathcal{UST}^{-}(q, \gamma).$ Thus for $\left \vert z\right \vert = r < 1,$ the following inequality is true:

Equality in (3.20) is attained for the function $f$ given by the formula

Proof. Since $f\in k-\mathcal{UST}^{-}(q, \gamma),$ in view of Theorem 4 we find

This gives

Therefore

and

The required results follows by letting $r\rightarrow 1^{-}.$ Hence the proof of Theorem 6 is complete.

Theorem 7. Let $k\in \lbrack 0, \infty),$ $q\in (0, 1),$ and $\gamma \in \mathbb{C} \backslash \{0\}.$ Let $\ f$ defined by (1.18) belongs to the class $k- \mathcal{UST}^{-}(q, \gamma).$ Thus, for $\left \vert z\right \vert = r < 1,$ the following inequality is true:

Proof. Differentiating $f$ and using triangle inequality for the modulus, we obtain

and

Assertion (3.23) follows from (3.24) and (3.25) in view of rather simple consequence of (3.22) given by the inequality

Hence we complete the proof of Theorem 7.

Theorem 8. The class $k-\mathcal{UST}^{-}(q, \gamma)$ is closed under convex linear combination.

Proof. Let the functions $f(z)$ and $g(z)$ are in class $k-\mathcal{UST} ^{-}(q, \gamma).$ Suppose $\ f(z)$ is given by (1.18) and

where $a_{n},$ $d_{n}\geq 0.$

It is sufficient to prove that for $0\leq \lambda \leq 1$, the function

is also in the class $k-\mathcal{UST}^{-}(q, \gamma).$

From (1.18), ($3.26$) and (3.27), we have

As $f(z)$ and $g(z)$ are in class $k-\mathcal{UST}^{-}(q, \gamma)$ and $0\leq \lambda \leq 1$, so by using Theorem 4, we obtain

Again by Theorem 4 and inequality (3.29), we have $H(z)\in k- \mathcal{UST}^{-}(q, \gamma).$ Hence the proof of Theorem 8 is complete.

4.   Conclusions

In this paper, motivated significantly by a number of recent works, we have made use of a certain general conic domain $\Omega _{k, q, \gamma }$ and the quantum (or $q$-) calculus in order to define and investigate a new subclass of normalized analytic functions in the open unit disk $E$ and we have successfully derived several properties and characteristics of newly defined subclass of analytic functions. For verification and validity of our main results we have also pointed out relevant connections of our main results with those in several earlier related works on this subject.

For further investigation, we can make obvious connections between the $q$ -analysis and $(p, q)$-analysis and the results for $q$-analogues which we have consider in this article for $0 < q < 1,$ can easily be translated into the corresponding results for the $(p, q)$-analogues with $(0 < q < p\leq 1)$ by applying some obvious parameter and argument variations.

Acknowledgments

This work was supported by the Natural Science Foundation of the Peoples Republic of China (Grant No. 11561001), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-18-A14), the Natural Science Foundation of Inner Mongolia of the Peoples Republic of China (Grant No. 2018MS01026), the Higher School Foundation of Inner Mongolia of the Peoples Republic of China (Grant No. NJZY18217) and the Natural Science Foundation of Chifeng of Inner Mongolia.

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.