Differential subordination, superordination results associated with Pascal distribution

1.   Introduction

Let $\mho$ be the family of holomorphic functions in $\Delta = \{z:|z| < 1 \}$ and $\mho[t, n]$ be the subclass of $\mho$ involving the functions which can be defined by

let $\mathcal{Q}$ be the subclass of $\mho$ involving the function defined by

Let ${\beta, \mathfrak{h}} \in \mho$ and consider $\theta(u, v, w, z):c^3 \times\Delta\rightarrow C$. If $\beta$ and $\theta(\beta(z), z\beta'(z), z^2\beta''(z), z)$ are univalent and if $\beta$ satisfies the second order superordination,

then $\beta$ is a solution of the differential subordination ^[2]. (If $g$ is subordinate to $G$, then $G$ is superordinate to $g$). An holomorphic function $\alpha$ is called a subordinate if $\alpha \prec \beta$ for every $\beta$ satisfying ^[2]. A univalent subordinant $\overline{\alpha}$ that satisfies $\alpha \prec \overline{\alpha}$ for all subordinates ^[2] $\alpha(z)$ is called the best subordinant.

Miller and Mocanu ^[5] found the conditions on $\mathfrak{h}, \alpha$ and $\theta$ it can be given by

For two holomorphic functions

The Hadamard product (or) convolution of $\lambda$ and $\mu$ given below

A variable $X$ is said to have the Pascal distribution series if it takes the values 0, 1, 2, 3... with the probabilities

respectively where $\alpha$, $r$ are called the parameters and thus

Many essentially interesting proof techniques involving a power series, whose co-efficients are probabilities of the Pascal distribution series introduced by sheeza et al. ^[12] that is

The first order differential subordination and superordination which was introduced and studied by Miller, Mocanu and Bulboaca ^[1,2,5]. Also recently studied by various authors for example Magesh and Murugusundaramoorthy ^[3,7,8,9], Magesh et al. ^[4], and Shanmugam et al. ^[11] and also obtained sandwich results for various classes of holomorphic functions.

In the present article we determine some sufficient condition for the holomorphic function in $\Delta$ to satisfy

where $\alpha_1, \alpha_2$ are given univalent functions in $\Delta$ with $\alpha_1(0) = 1, \alpha_2(0) = 1.$

2.   Preliminary results

To prove our results we need the following lemmas and definitions.

Lemma 2.1. ^[10] The function

is a subordination chain, if

Definition 2.2. ^[5] Denote by $T$, the set of all functions $f$ that are holomorphic and one to one on $\overline{\Delta} - E(f)$ where,

and are such that $f'(\zeta)\neq 0$, for $\zeta \in \partial\Delta-E(f)$.

Lemma 2.3. ^[6] Let $\alpha$ be univalent in the unit disc $\Delta$ and $\psi$ and $\theta$ be holomorphic in a domain $D$ containing $\alpha(\Delta)$ with $\theta(\omega) \neq 0$ when $\omega \in \alpha (\Delta)$. Set

suppose that

(1) T(z) is starlike univalent in $\Delta$.

(2) $\mathfrak{R}\left\{ \frac{z\mathfrak{h}'(z)}{T(z)}\right\} > 0$, for $z \in \Delta.$

If $\beta$ is holomorphic with $\alpha(0) = \beta(0)$, $\beta(\Delta) \subseteq D,$ and

then

and $\alpha$ is th best dominant.

Lemma 2.4. ^[2] Let $\alpha$ be convex univalent in the unit disk $\Delta$ and $\upsilon$ and $\varrho$ be holomorphic in a domain $D$ containing $\alpha(\Delta)$. Suppose that

(1) $\mathfrak{R}\left\{\frac{ \upsilon'(\alpha(z))}{\varrho(\alpha(z))}\right\} > 0$, for $z \in \Delta.$

(2) $\phi(z) = z\alpha'(z)\varrho(\alpha(z))$ is starlike univalent in $\Delta$.

If $\beta(z) \in \mho[\alpha(0), 1]\cap T$ with $\beta(\Delta) \subseteq D$ and $\upsilon(\alpha(z))+z\beta'(z)\varrho(\beta(z))$ is univalent in $\Delta$ and

then

and $\alpha$ is the best subordinant.

3.   Subordination results

To prove our following theorem need to using above Lemma 2.3.

Theorem 3.1. Let $Q_\alpha^r \in Q$, $\eta_i \in C (i = 1, 2, 3), \; (\eta_3 \neq 0), \; \wp \in C,$ such that $\wp \neq 0$, $\alpha$ be convex univalent with $\alpha(0) = 1,$ and assume that

which is greater than zero, $z \in \Delta.$

If $g \in Q$ satisfies

where

then

and $\alpha$ is the best dominant.

Proof. Define the function $\beta$ by

then the function $\beta$ is holomorphic in $\Delta$ and $\beta(0) = 1.$ Therfore, by making use of (3.4), we obtain

by using (3.5) in (3.2), we have

By setting

and

This is easily observed that $\psi(\alpha(z))$, $\theta(z)$ are holomorphic in $c-\{0\}$ and $\theta(z) \neq 0.$ Also we see that

and

Here T(z) is starlike univalent in $\Delta$ and we get the result

Hence the theorem.

By taking

in Theorem 4.1, we obtain the following corollary.

Corollary 3.2. Let $\eta_i \in C \; \; (i = 1, 2, 3), \; (\eta_3 \neq 0), \; \wp \in C, \; s.t\; \wp \neq 0$ be convex univalent with $\alpha(0) = 1$ and (3.1) hold true. For $g, Q_\alpha^r \in Q,$ let $\left(\frac{Q_\alpha^r}{z}\right)^\wp \in H[1,1] \cap T$ and $\nabla(\eta_i)_1^3(g; Q_\alpha^r)$ defined in (3.3) be univalent in $\Delta$ satisfying

then

and $\frac{1+Az}{1+Bz}$ is the best dominant.

4.   Superordination results

Using Lemma 2.4 to prove the following theorem.

Theorem 4.1. Let $Q_\alpha^r(z) \in Q$, $\eta_i \in C (i = 1, 2, 3), \; (\eta_3 \neq 0), \; \wp \in C, \;$s.t $\wp \neq 0$, $\alpha$ be convex univalent with $\alpha(0) = 1$, and assume that

If $g \in Q$, $Q_\alpha^r(z) \in H, \; [\alpha(0), 1] \bigcap T$, Let $\nabla^{(\eta_i)_1^3}(g; Q_\alpha^r)$ be univalent in $\Delta$ and

where $\nabla^{(\eta_i)_1^3}(g; Q_\alpha^r)$ is given in (3.3), then

and $\alpha$ is the best subordinant.

Proof. The function $\beta$ is defined by

simplify above equation, we get

then

By setting

Here $\upsilon(\alpha(z))$ is holomorphic in $C$. Also $\varrho(z)$ is holomorphic in $C-\{0\}$ and $\varrho(z) \neq 0.$ Consider,

differentiating the above equation with respect to $z$ and $n$, we have

and

From the univalence of $\alpha$ we have $\alpha'(0) \neq 0$ and $\alpha(0) = 1,$ it follows that $t_1(n) \neq 0$ for $n \geq 0$ and $\lim_{n \to \infty} |t_1(n)| = \infty.$

A simple compution yields,

Using the fact that $\alpha$ is convex univalent function in $\Delta$ and $\eta_3 \neq 0$ we have

Hence the theorem.

By taking

in Theorem 4.1 we obtain the following corollary.

Corollary 4.2. Let $\eta_i \in C\; (i = 1, 2, 3), \; (\eta_3 \neq 0), \; \wp \in C, \; s.t\; \wp \neq 0$ be convex univalent with $\alpha(0) = 1$ and (4.1) hold true. For $g, Q_\alpha^r \in Q,$ let $\left(\frac{Q_\alpha^r}{z}\right)^\wp \in H[1,1] \cap T$ and $\nabla(\eta_i)_1^3(g; Q_\alpha^r)$ defined in (3.3) be univalent in $\Delta$ satisfying

then

and $\frac{1+Az}{1+Bz}$ is the best subordinant.

5.   Sandwich theorem

To obtain the sandwich results get from combining the subordination results and superordination results

Theorem 5.1. Let $\alpha_1$ and $\alpha_2$ be convex univalent in $\Delta$, $\eta_i \in C\; (i = 1, 2, 3), \; (\eta_3 \neq 0), \; \wp \in C, \; s.t\; \wp \neq 0$ and let $\alpha_2$ satisfying (3.1) and $\alpha_1$ satisfying (4.1). For $g, Q_\alpha^r \in Q,$ let $\left(\frac{Q_\alpha^r}{z}\right)^\wp \in H[1,1] \cap T$ and $\nabla(\eta_i)_1^3(g; Q_\alpha^r)$ defined in (3.3) be univalent in $\Delta$ satisfying

then

and $\alpha_1$, $\alpha_2$ are respectively best subordinant and best dominant.

Hence the proof of the theorem. By taking

and

in Theorem 5.1, we obtain the following result.

Corollary 5.2. For $g, Q_\alpha^r \in Q,$ let $\left(\frac{Q_\alpha^r}{z}\right)^\wp \in H[1,1] \cap T$ and $\nabla(\eta_i)_1^3(g; Q_\alpha^r)$ defined in (3.3) be univalent in $\Delta$ satisfying

then

and $\frac{1+A_1z}{1+B_1z}$, $\frac{1+A_2z}{1+B_2z}$ are respectively the best subordinant and best dominant.

6.   Conclusions

This paper deals with the applications of the differential subordination and superordination results involving Pascal distribution series. In addition we found the sandwich results to be in the class of holomorphic functions. Many interesting particular cases of the main theorems are emphazied in the form of corollaries. Furthermore to illustrate the results of application in various classes of analytic function. We anticipate that differential subordination and superordination will be important in several fields related to mathematics, science and technology.

Acknowledgments

The authors would like to thanks the reviewers for the deep comments to improve our work. Also, we express our thanks to editorial office for their advice.

Conflict of interest

The authors declare no conflicts of interest.