Generalized hypergeometric distribution and its applications on univalent functions

Abstract

The purpose of the present paper is to introduce a generalized hypergeometric distribution and obtain some necessary and sufficient conditions for generalized hypergeometric distribution series belonging to certain classes of univalent functions associated with the conic domains. We also investigate some inclusion relations. Finally, we discuss an integral operator related to this series.

1 Introduction

Let $\mathscr{A}$ denote the class of functions $f(z)$ of the form

$$ f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n} $$

(1.1)

that are analytic in the open unit disk

$$ \mathbb{U}:= \bigl\{ z \in \mathbb{C}~:~ \vert z \vert < 1 \bigr\} $$

and satisfy the normalized conditions $f(0)=f^{\prime }(0)-1=0$. As usual, we denote by $\mathscr{S}$ the subclass of $\mathscr{A}$ consisting of functions of the form (1.1) that are also univalent in $\mathbb{U}$. Further, by $\mathscr{T}$ we denote the subclass of $\mathscr{S}$ consisting of functions of the form

$$ f(z)=z-\sum_{n=2}^{\infty } \vert a_{n} \vert ~z^{n}. $$

(1.2)

A function $f\in \mathscr{A}$ is said to be a starlike function of order α ($0 \leq \alpha < 1$) if

$$\begin{aligned} \Re \biggl(\frac{zf'(z)}{f(z)} \biggr)>\alpha , \quad z\in \mathbb{U}. \end{aligned}$$

We denote the class of starlike functions of order α by $\mathscr{S}^{\ast }(\alpha )$.

A function $f\in \mathscr{A}$ is said to be a convex function of order α ($0 \leq \alpha < 1$) if

$$\begin{aligned} \Re \biggl(1+\frac{zf''(z)}{f'(z)} \biggr)>\alpha , \quad z\in \mathbb{U}. \end{aligned}$$

We denote the class of convex functions of order α by $\mathscr{K}(\alpha )$.

Further, let $\mathscr{S}^{\ast }(0)\equiv \mathscr{S}^{\ast }$ and $\mathscr{K}(0) \equiv \mathscr{K}$ be the well-known standard classes of starlike and convex functions. The classes of starlike and convex functions of order α were studied earlier by Robertson [17] and Silverman [18].

For some α ($0 \leq \alpha < 1$) and $\beta \geq 0$ and functions of the form (1.1), let $\mathscr{S}_{p}(\alpha , \beta )$ be the subclass of $\mathscr{S}$ satisfying the analytic criteria

$$\begin{aligned} \Re \biggl(\frac{zf'(z)}{f(z)} \biggr)-\alpha > \beta \biggl\vert \frac{zf'(z)}{f(z)}-1 \biggr\vert , \quad z\in \mathbb{U}, \end{aligned}$$

(1.3)

and let $\mathscr{UCV}(\alpha , \beta )$ be the subclass of $\mathscr{S}$ satisfying the analytic criteria

$$\begin{aligned} \Re \biggl(1+\frac{zf''(z)}{f'(z)} \biggr) > \beta \biggl\vert \frac{zf''(z)}{f'(z)} \biggr\vert + \alpha , \quad z\in \mathbb{U}. \end{aligned}$$

(1.4)

Note that $\mathscr{S}_{p}(\alpha , \beta )\cap \mathscr{T}=\mathscr{TS}_{p}( \alpha , \beta )$ and $\mathscr{UCV}(\alpha , \beta )\cap \mathscr{T}=\mathscr{TUCV}( \alpha , \beta )$.

The classes $\mathscr{S}_{p}(\alpha , \beta )$, $\mathscr{UCV}(\alpha , \beta )$, $\mathscr{TS}_{p}(\alpha , \beta )$, and $\mathscr{TUCV}(\alpha , \beta )$ were studied by Bharati et al. [3].

By specializing the parameters in $\mathscr{S}_{p}(\alpha , \beta )$ and $\mathscr{UCV}(\alpha , \beta )$ we obtain following known subclasses studied earlier by various researchers:

• (1) $\mathscr{S}_{p}(0, \beta ) \equiv \mathscr{S}_{p}(\beta )$ and $\mathscr{UCV}(0, \beta ) \equiv \mathscr{UCV}(\beta )$ studied by Kanas and Wisniowska [8, 9].

• (2) $\mathscr{S}_{p}(0, 1) \equiv \mathscr{S}_{p}$ and $\mathscr{UCV}(0, 1) \equiv \mathscr{UCV}$ studied by Goodman [6, 7] (see also [10]).

• (3) $\mathscr{S}_{p}(\alpha , 0) \equiv \mathscr{S}^{\ast }(\alpha )$, $\mathscr{UCV}(\alpha , 0) \equiv \mathscr{K}(\alpha )$, $\mathscr{S}_{p}(0, 0) \equiv \mathscr{S}^{\ast }$, and $\mathscr{UCV}(0, 0) \equiv \mathscr{K}$ studied by Robertson [17] and Silverman [18].

In 1995, Dixit and Pal [4] introduced the class $\mathscr{R}^{\tau } (A, B )$ consisting of functions $f(z)$ of the form (1.1) that satisfy the inequality

$$ \biggl\vert \frac{f^{\prime }(z) - 1}{(A-B)\tau - B(f^{\prime }(z) - 1)} \biggr\vert < 1, \quad \tau \in \mathbb{C} \setminus \{0\}, \quad -1 \leq B < A \leq 1, \quad z \in \mathbb{U}. $$

For complex numbers $a_{1}, a_{2}, \dots ,a_{p}$ and $b_{1}, b_{2}, \dots , b_{q}$ with $b_{j}\neq 0,-1,-2, \dots ,j = 1,2,\dots ,q$, the generalized hypergeometric functions $_{p}F_{q}(a_{1}, a_{2}, \ldots a_{p}; b_{1}, b_{2}, \ldots b_{q}; z)$ are defined by

$$ _{p}F_{q}(a_{1}, a_{2}, \ldots a_{p}; b_{1}, b_{2}, \ldots b_{q}; z) = \sum _{n=0}^{\infty } \frac{(a_{1})_{n} \cdots (a_{p})_{n}}{(b_{1})_{n} \cdots (b_{q})_{n}} \frac{z^{n}}{n!}, \quad z \in \mathbb{U}, $$

(1.5)

where $p \leq q + 1$, and $(a)_{n}$ is the Pochhammer symbol defined by

$$ (a)_{n} = \frac{\Gamma (a + n)}{\Gamma (a)} = \textstyle\begin{cases} 1 & \text{if } n = 0, \\ a (a +1)(a +2)\dots (a +n-1) & \text{if } n \in \mathbb{N}. \end{cases} $$

(1.6)

The convergence conditions for the series defined by (1.5) are as follows:

• (1) If $p < q + 1$, then the series converges absolutely in the entire complex plane.

• (2) If $p \leq q$, then the series converges absolutely for every finite z.

• (3) If $p = q + 1$, then the series converges absolutely for $|z| < 1$.

• (4) If $p = q + 1$ and $|z|=1$, then the series converges when $\Re \{ \sum_{j = 1}^{q}b_{j} - \sum_{i = 1}^{p}a_{i} \} > 0$.

For a detailed study, we refer to [16].

Now for $a_{i}$, $i = 1, 2,\dots ,p, b_{j}, j = 1, 2,\dots , q$, and $m > 0$, we define

$$ _{p}F_{q}(a_{1}, a_{2}, \ldots a_{p}; b_{1}, b_{2}, \ldots b_{q}; m) = \sum\limits _{n=0}^{\infty } \frac{(a_{1})_{n} \cdots (a_{p})_{n}}{(b_{1})_{n} \cdots (b_{q})_{n}} \frac{m^{n}}{n!}, $$

provided that the series is convergent.

In this paper, we use the notations

$$ _{p}F_{q} (z) = _{p}F_{q}(a_{1}, a_{2}, \ldots a_{p}; b_{1}, b_{2}, \ldots b_{q}; z) $$

and

$$ _{p}F_{q}(a_{1}+k; b_{1}+k; z) = _{p}F_{q}(a_{1}+k, a_{2}+k, \ldots a_{p}+k; b_{1}+k, b_{2}+k, \ldots b_{q}+k; z),\quad k \in \mathbb{N}. $$

Now we introduce the generalized hypergeometric distribution with probability mass function

$$ \frac{(a_{1})_{n} \cdots (a_{p})_{n}}{(b_{1})_{n} \cdots (b_{q})_{n}} \frac{m^{n}}{n!} \frac{1}{_{p}F_{q}(m)},\quad n = 0, 1, 2, \dots . $$

By specializing the parameters in the generalized hypergeometric distribution it reduces to the following probability distributions:

• (1) If $p=2$ and $q=1$, then it reduces to the hypergeometric-type probability distribution studied by Porwal and Gupta [15].

• (2) If $p=q=1$, then it reduces to the confluent hypergeometric distribution studied by Porwal [14].

• (3) If $p=q=1$ and $a_{1} = b_{1}$, then it reduces to the well-known Poisson distribution.

Next, we introduce the generalized hypergeometric distribution series whose coefficients are the probabilities of generalized hypergeometric distribution

$$ {}_{p}F_{q}(m, z) = z + \sum _{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{z^{n}}{_{p}F_{q}(m)}, $$

(1.7)

where $a_{i}, b_{j} > 0$, $i = 1, 2, \ldots , p$, $j = 1, 2, \ldots , q$.

Now we define

$$ {}_{p}\overline{F}_{q}(m, z) = 2z - {}_{p}F_{q}(m, z) = z - \sum _{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{z^{n}}{_{p}F_{q}(m)}. $$

(1.8)

The convolution (or Hadamard product) of two power series is defined as

$$ (f \ast g) (z) = f(z) \ast g(z) = \sum\limits _{n=0}^{\infty } a_{n}b_{n}z^{n}, \quad z \in \mathbb{U}. $$

Now we consider the linear operator $\Omega (p, q, m) : \mathscr{A} \rightarrow \mathscr{A}$ defined by

$$\begin{aligned} \Omega (p, q; m) f(z) = {}_{p}F_{q}(m, z) \ast f(z) = z + \sum_{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{a_{n}z^{n}}{_{p}F_{q}(m)}. \end{aligned}$$

(1.9)

In 2014, Porwal [12] introduced the Poisson distribution series and obtained necessary and sufficient conditions for this series to belong to certain classes of univalent functions. It opens up a new and interesting direction of research in geometric function theory. After the appearance of this paper, several researchers introduced the hypergeometric distribution series [1], the binomial distribution series [11], the hypergeometric-type distribution series [15], the confluent hypergeometric distribution series [14], the Pascal distribution series [5], the generalized distribution series [13], and the Mittag-Leffler-type Poisson distribution series [2] and obtained some necessary and sufficient conditions for them to belong to certain classes of univalent functions. Motivated with the works mentioned, in this paper, we obtain some necessary and sufficient conditions for the generalized hypergeometric distribution series to belong to the classes $\mathscr{S}_{p}(\alpha , \beta )$, $\mathscr{UCV}(\alpha , \beta )$, and $\mathscr{UCV}(\beta )$. We also obtain some inclusion relations between the classes $\mathscr{R}^{\tau }(A, B)$ and $\mathscr{UCV}(\alpha , \beta )$, $\mathscr{UCV}(\beta )$. Finally, we discuss an integral operator associated with the generalized distribution series.

2 Main results

To prove our main results, we need the following lemmas.

Lemma 2.1

([13])

A function $f \in \mathscr{A}$ of the form (1.1) belongs to the class $\mathscr{S}_{p}(\alpha , \beta )$ if

$$ \sum_{n=2}^{\infty } \bigl[n(1+\beta ) - (\alpha + \beta ) \bigr] \vert a_{n} \vert \leq 1 - \alpha . $$

(2.1)

Lemma 2.2

([3])

A function $f \in \mathscr{A}$ of the form (1.1) is said to be in the class $\mathscr{UCV}(\alpha , \beta )$ if

$$ \sum_{n=2}^{\infty } n \bigl[n(1+\beta ) - (\alpha + \beta ) \bigr] \vert a_{n} \vert \leq 1 - \alpha . $$

(2.2)

Remark 2.1

Conditions (2.1) and (2.2) are also necessary for functions $f(z)$ of the form (1.2).

Lemma 2.3

([8])

A function $f \in \mathscr{A}$ of the form (1.1) is said to be in the class $\mathscr{UCV}(\beta )$ if

$$ \sum_{n=2}^{\infty } n(n-1) \vert a_{n} \vert \leq \frac{1}{\beta + 2}. $$

(2.3)

The number $\frac{1}{\beta + 2}$ cannot be increased.

Lemma 2.4

([4])

If $f \in \mathscr{R}^{\tau } (A, B )$ is of the form (1.1), then

$$ \vert a_{n} \vert \leq \frac{(A-B) \vert \tau \vert }{n}, \quad n \in \mathbb{N} \setminus \{1\}. $$

(2.4)

The bounds given in (2.4) are sharp.

Theorem 2.1

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$. Suppose that the inequality

$$\begin{aligned}& (1 + \beta ) \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p} + 1)}{b_{1}(b_{1} + 1) \cdots b_{q}(b_{q} + 1)} m^{2} {}_{p}F_{q}(a_{1} + 2; b_{1} + 2; m) \\& \quad{} + (3 + 2 \beta - \alpha ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \leq 1 - \alpha \end{aligned}$$

(2.5)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 2$.

Then ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{UCV}(\alpha , \beta )$.

Proof

To prove that ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{UCV}(\alpha , \beta )$, by Lemma 2.2 it suffices to prove that

$$ \sum\limits _{n=2}^{\infty } n \bigl[n (1 + \beta ) - ( \alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \leq 1 - \alpha . $$

We have

$$\begin{aligned}& \sum_{n=2}^{\infty } n \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ \sum_{n=2}^{\infty } \bigl[(n - 1) (n - 2) (1 + \beta ) + (3 + 2 \beta - \alpha ) (n - 1)\\& \qquad{} + (1 - \alpha ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \sum _{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-3)!} \\& \qquad{}+ (3 + 2 \beta - \alpha ) \sum_{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-2)!} \\& \qquad{}+ (1 - \alpha ) \sum_{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1) \cdots b_{q}(b_{q}+1)} m^{2} \sum_{n=2}^{\infty } \frac{(a_{1} + 2)_{n-3} \cdots (a_{p} + 2)_{n-3}}{(b_{1} + 2)_{n-3} \cdots (b_{q} + 2)_{n-3}} \frac{m^{n-3}}{(n-3)!} \\& \qquad {}+ (3 + 2 \beta - \alpha ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m \sum _{n=2}^{ \infty } \frac{(a_{1} + 1)_{n-2} \cdots (a_{p} + 1)_{n-2}}{(b_{1} + 1)_{n-2} \cdots (b_{q} + 1)_{n-2}} \frac{m^{n-2}}{(n-2)!} \\& \qquad{} + (1 - \alpha ) \sum_{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \biggl[ (1 + \beta ) \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1) \cdots b_{q}(b_{q}+1)} m^{2} {}_{p}F_{q}(a_{1} + 2; b_{1} + 2; m) \\& \qquad{} + (3 + 2 \beta - \alpha ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) + (1 - \alpha ) \bigl({}_{p}F_{q}(m) - 1 \bigr) \biggr] \\& \quad \leq 1 - \alpha \end{aligned}$$

by the given hypothesis. This completes the proof of Theorem 2.1. □

Theorem 2.2

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$. Suppose that the inequality

$$\begin{aligned} (1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \leq 1 - \alpha \end{aligned}$$

(2.6)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 1$.

Then ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{S}_{p}(\alpha , \beta )$.

Proof

To prove that ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{S}_{p}(\alpha , \beta )$, by Lemma 2.1 it suffices to prove that

$$ \sum_{n=2}^{\infty } \bigl[n (1 + \beta ) - ( \alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \leq 1 - \alpha . $$

We have

$$\begin{aligned}& \sum_{n=2}^{\infty } \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \sum _{n=2}^{ \infty } (n - 1) \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!}\\& \qquad{} + (1 - \alpha ) \sum_{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m \sum _{n=2}^{ \infty } \frac{(a_{1} + 1)_{n-2} \cdots (a_{p} + 1)_{n-2}}{(b_{1} + 1)_{n-2} \cdots (b_{q} + 1)_{n-2}} \frac{m^{n-2}}{(n-2)!} \\& \qquad{} + (1 - \alpha ) \sum_{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \biggl[ (1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) + (1 - \alpha ) \bigl({}_{p}F_{q}(m) - 1 \bigr) \biggr] \\& \quad \leq 1 - \alpha , \end{aligned}$$

by the given hypothesis. This completes the proof of Theorem 2.2. □

Remark 2.2

Conditions (2.5) and (2.6) are also necessary for the series ${}_{p}\overline{F}_{q}(m, z)$ defined by (1.8) to belong to the classes $\mathscr{TS}_{p}(\alpha , \beta )$ and $\mathscr{TUCV}(\alpha , \beta )$, respectively.

Theorem 2.3

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$. Suppose that the inequality

$$\begin{aligned}& \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p} + 1)}{b_{1}(b_{1} + 1) \cdots b_{q}(b_{q} + 1)} m^{2} {}_{p}F_{q}(a_{1} + 2; b_{1} + 2; m) + 2 \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \\& \quad \leq \frac{{}_{p}F_{q}(m)}{\beta + 2} \end{aligned}$$

(2.7)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 2$.

Then ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{UCV}(\beta )$.

Proof

To prove that ${}_{p}F_{q}(m, z)$ defined by (1.7) is in the class $\mathscr{UCV}(\beta )$, by Lemma 2.3 it suffices to prove that

$$ \sum_{n=2}^{\infty } n [n - 1] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \leq \frac{1}{\beta + 2}. $$

We have

$$\begin{aligned}& \sum_{n=2}^{\infty } n [n - 1] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{1}{{}_{p}F_{q}(m)} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ \sum_{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-3)!} + 2 \sum _{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-2)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1) \cdots b_{q}(b_{q}+1)} m^{2} \sum _{n=2}^{\infty } \frac{(a_{1} + 2)_{n-3} \cdots (a_{p} + 2)_{n-3}}{(b_{1} + 2)_{n-3} \cdots (b_{q} + 2)_{n-3}} \frac{m^{n-3}}{(n-3)!} \\& \qquad{} + 2 \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m \sum_{n=2}^{ \infty } \frac{(a_{1} + 1)_{n-2} \cdots (a_{p} + 1)_{n-2}}{(b_{1} + 1)_{n-2} \cdots (b_{q} + 1)_{n-2}} \frac{m^{n-2}}{(n-2)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \biggl[ \frac{a_{1}(a_{1}+1) \cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1) \cdots b_{q}(b_{q}+1)} m^{2} {}_{p}F_{q}(a_{1} \\& \qquad{} + 2; b_{1} + 2; m) + 2 \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \biggr] \\& \quad \leq \frac{1}{\beta + 2} \end{aligned}$$

by the given hypothesis. This completes the proof of Theorem 2.3. □

Theorem 2.4

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$, and let $f \in \mathscr{R}^{\tau } (A, B )$. Suppose that the inequality

$$\begin{aligned}& \frac{(A - B) \vert \tau \vert }{{}_{p}F_{q}(m) } \biggl[(1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) + (1 - \alpha ) \bigl({}_{p}F_{q}(m) - 1 \bigr) \biggr] \\& \quad \leq 1 - \alpha \end{aligned}$$

(2.8)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 1$.

Then $\Omega (p, q, m) f \in \mathscr{UCV}(\alpha , \beta )$.

Proof

Since

$$ \Omega (p, q, m) f(z) = z + \sum\limits _{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{a_{n}z^{n}}{_{p}F_{q}(m)}, $$

to prove that $\Omega (p, q, m) f \in \mathscr{UCV}(\alpha , \beta )$, by Lemma 2.2 it suffices to prove that

$$\begin{aligned} \sum_{n=2}^{\infty } n \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{ \vert a_{n} \vert }{{}_{p}F_{q}(m)} \leq 1 - \alpha . \end{aligned}$$

Using Lemma 2.4, we have

$$\begin{aligned}& \sum_{n=2}^{\infty } n \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \frac{ \vert a_{n} \vert }{{}_{p}F_{q}(m)} \\& \quad \leq \frac{(A - B) \vert \tau \vert }{{}_{p}F_{q}(m)} \sum_{n=2}^{ \infty } \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \\& \quad = \frac{(A - B) \vert \tau \vert }{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \sum _{n=2}^{\infty } (n - 1) \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \\& \qquad{} + (1 - \alpha ) \sum_{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{(A - B) \vert \tau \vert }{{}_{p}F_{q}(m)} \biggl[ (1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1, b_{1} + 1, m) + (1 - \alpha ) \bigl({}_{p}F_{q}(m) - 1 \bigr) \biggr] \\& \quad \leq 1 - \alpha \end{aligned}$$

by the given hypothesis. This completes the proof of Theorem 2.4. □

Theorem 2.5

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$, and let $f \in \mathscr{R}^{\tau } (A, B )$. Suppose that the inequality

$$\begin{aligned} \frac{(A - B) \vert \tau \vert }{{}_{p}F_{q}(m)} \biggl[ \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \biggr] \leq \frac{1}{\beta + 2} \end{aligned}$$

(2.9)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 1$.

Then $\Omega (p, q, m) f \in \mathscr{UCV}(\beta )$.

Proof

The proof of Theorem 2.5 is similar to that of Theorem 2.4, and therefore we omit it. □

Remark 2.3

If we put $p = q = 1$ in Theorems 2.1 and 2.2, then we obtain the corresponding results of Porwal [14].

3 An integral operator

In this section, we obtain analogous results in connection with the particular integral

$$ G(p, q, m, z) = \int _{0}^{z} \frac{{}_{p}F_{q} (m, t)}{t} \,dt. $$

(3.1)

Theorem 3.1

If all the conditions of Theorem 2.2hold, then $G(p, q, m, z)$ defined by (3.1) is in the class $\mathscr{UCV}(\alpha , \beta )$.

Proof

From representation (3.1) we have

$$ G(p, q, m, z) = z + \sum\limits _{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{n!} \frac{z^{n}}{_{p}F_{q}(m)}. $$

To prove that $G(p, q, m, z) \in \mathscr{UCV}(\alpha , \beta )$, by Lemma 2.2 it suffices to prove that

$$\begin{aligned} \sum_{n=2}^{\infty } n \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{n!} \frac{1}{{}_{p}F_{q}(m)} \leq 1 - \alpha . \end{aligned}$$

We have

$$\begin{aligned}& \sum_{n=2}^{\infty } n \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{n!} \frac{1}{{}_{p}F_{q}(m)} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \sum_{n=2}^{\infty } \bigl[n (1 + \beta ) - (\alpha + \beta ) \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n - 1)!} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \sum_{n=2}^{\infty } \bigl[(1 + \beta ) (n - 1) + 1 - \alpha \bigr] \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n - 1)!} \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \Biggl[ (1 + \beta ) \sum _{n=2}^{ \infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-2)!} \\& \qquad {}+ (1 - \alpha ) \sum_{n=2}^{\infty } \frac{(a_{1})_{n-1} \cdots (a_{p})_{n-1}}{(b_{1})_{n-1} \cdots (b_{q})_{n-1}} \frac{m^{n-1}}{(n-1)!} \Biggr] \\& \quad = \frac{1}{{}_{p}F_{q}(m)} \biggl[ (1 + \beta ) \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) + (1 - \alpha ) \bigl({}_{p}F_{q}(m) - 1 \bigr) \biggr]. \end{aligned}$$

The last expression is bounded above by $1 - \alpha$ if (2.6) holds. Thus the proof of Theorem 3.1 is established. □

Theorem 3.2

Let $a_{i}, b_{j} > 0$ $(i = 1, 2, \ldots , p; j = 1, 2, \ldots , q)$. Suppose that the inequality

$$\begin{aligned} \frac{a_{1} \cdots a_{p}}{b_{1} \cdots b_{q}} m {}_{p}F_{q}(a_{1} + 1; b_{1} + 1; m) \leq \frac{{}_{p}F_{q}(m)}{\beta + 2} \end{aligned}$$

(3.2)

holds with one of the following conditions:

• (1) $p \leq q$ and $m > 0$,

• (2) $p = q + 1$ and $m < 1$,

• (3) $p = q + 1$, $m = 1$, and $\sum_{j = 1}^{q} b_{j} > \sum_{i = 1}^{p} a_{i} + 1$.

Then $G(p, q, m, z)$ defined by (3.1) is in the class $\mathscr{UCV}(\beta )$.

Proof

The proof of this theorem is much akin to that of Theorem 3.1. Therefore we omit it. □

Remark 3.1

If we put $p=2$ and $q=1$ in Theorems 2.1–3.2, then we obtain the corresponding results of Porwal and Gupta [15].