A New Subclass of Multivalent Function Defined by Using Jackson Derivative Operator

Shivani Indora; S.K.Bissu

I. INTRODUCTION

Let $\mathcal{H}(p)$ be the class of analytic and $p$ -valent function $f(z)$ . The function $f(z)$ can be expressed as

where $p$ is some natural number, $n \in \mathbb{N}$

The function $f(z)$ defined in (1.1) is an analytic function and $p$ -valent function in the open unit disc

If a function $f(z) \in \mathcal{H}(p)$ satisfies the following condition

then $f(z)$ is a $p$ -valent starlike function of order $\varphi$ .

and if a function $f(z) \in \mathcal{H}(p)$ satisfies the following condition

then $f(z)$ is a $p$ -valent convex function of order $\varphi$

To define a new subclass of multivalent function by using Jackson derivative, we use the following definitions

Definition 1: Let $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and $g(z) = z^{p} - \sum_{k=n+p}^{\infty} b_{k}z^{k}$ are the members of the class $\mathcal{H}(p)$ , then their convolution product or Hadamard product is defined as

and generally the convolution product of functions $f(z)$ and $g(z)$ is denoted by $(f*g)(z)$ or $(g*f)(z)$ .

Definition 2: The Jackson q-derivative of a function $f(z)$ is denoted by $D_q f(z)$ or $D_{q,z}f(z)$ and it is defined as

The Jackson's q - derivative tends to ordinary derivative when q tends to 1. The Jackson q- derivative can also be written as

A new class of multivalent function form by using Jackson Derivative Operator is defined in the following definition.

Definition 5.3: A function $f(z) \in \mathcal{H}(p)$ is also belongs to new subclass $\Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathbf{q})$ if it follows the following condition

where $z \in U_1, m \in \mathbb{N} \cup \{0\}, 0 \leq \alpha < p, 0 \leq \beta < 1,0 \leq \lambda \leq 1$ and $0 \leq \xi < 1$ .

By taking particular values of the parameters, $n,p,q,\beta,\lambda,\xi$ we get the previously defined subclasses of univalent and multivalent function. These classes were studied by Silverman [14], Srivastava [15], Altintas et. al [2] and Khosravianarb et. al [7].

Particular Cases:

If $m = 0, \beta = 0, \xi = 0, q \to 1$ then from (1.7) we get

which is equivalent to

sowe get $\Psi_{0,n,p}(\alpha,0,\lambda,0,1) \equiv T(n,p,\lambda,\alpha)$ and this class was studied by Altintas et al. [2]

If $m = 0, \beta = 0, q \to 1$ then from (1.7) we get

so $\Psi_{0,n,p}(\alpha,0,\lambda,\xi,1) \equiv T(n,p,\lambda,\alpha,\xi)$ and this class was studied by Khosravianarb et al.[7]

If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0$ then from (1.7) we get

so $\Psi_{0,n,p}(\alpha, 0,0,0,1) \equiv \mathrm{T}^{*}(p, \alpha)$ and $\mathrm{T}^{*}(p, \alpha)$ is the class of p valent starlike function of order $\alpha$ .

If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0, p = 1$ then from (1.7) we get

so $\Psi_{0,n,1}(\alpha, 0,0,0,1) \equiv \mathrm{T}^{*}(1, \alpha)$ , which was earlier studied by Srivastava et al. [15].

If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0, p = 1, n = 1$ then from (1.7) we get

Then we get a class which was earlier discussed by Silverman [14].

If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 1$ then from (1.7) we get

which is equivalent to $Re\left\{1 + \frac{zf''(z)}{f'(z)}\right\} > \alpha$ $0 \leq \alpha < p$

so $\Psi_{0,n,p}(\alpha, 0,1,0,1) \equiv \mathsf{C}^{*}(p, \alpha)$ and $\mathsf{C}^{*}(p, \alpha)$ represent a class of p valent convex function of order $\alpha$ .

If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 1, p = 1$ then from (1.7) we get

sowe get $\Psi_{0,n,1}(\alpha,0,1,0,1)\equiv \mathrm{C}^{*}(1,\alpha)$ ,which was earlier by studied Srivastava et al. [15]. 8.If $m = 0,\beta = 0,\xi = 0,q\to 1,\lambda = 1,p = 1,n = 1$ then from (1.7) we get

and this class of convex function was first introduced by Silverman [14].

II. COEFFICIENT ESTIMATE

In this part of the paper we derive the coefficient estimate of function $f(z)$ , $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathbf{q})$

Theorem 1: A function $\mathrm{f(z) = z^p - \sum_{k = n + p}^\infty a_kz^k}$ and $\mathrm{f(z)\in\mathcal{H}(p)}$ then $\mathrm{f(z)}$ belong to the class $\Psi_{\mathrm{m,n,p}}(\alpha,\beta,\lambda,\xi,\mathfrak{q})$ if and only if

Proof: Let us consider that $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ so we have

Since $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and

so we have

By using (2.2), (2.3) and (2.4) in (2.1) then we get numerator and denominator of (2.1) as numerator is denoted by $\mathbf{N}$ and denominator by $\mathbf{D}$

solve above by using $[n]_q = \frac{\Gamma_q(1 + n)}{\Gamma_q(n)}$ and on considering the value of $z$ to be real and let $z \to 1$ then we get

on simplifying we get,

Conversely: Let us assume the inequality (2.1) is true To Prove: $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ , for this we have to show that

According to Lemma [4] if $w = u + iv$ then $Re w \geq \alpha \Leftrightarrow |w - (1 + \alpha)| \leq |w + (1 - \alpha)|$ (2.5)

Let $L = |w - (1 + \alpha)|$ .

and $w = \frac{(z + \beta z^2)(D_{q,z}^{m + \xi + 1}f(z)) + \lambda(z^2 + \beta z)(D_{q,z}^{m + \xi + 2}f(z))}{(1 - \lambda)(D_{q,z}^{m + \xi}f(z)) + \lambda(z + \beta z^2)(D_{q,z}^{m + \xi + 1}f(z))}$ (2.6)

and $K = |w + (1 - \alpha)|$ .

From (2.7) and (2.8), $K - L > 0$

i.e. $|w + (1 - \beta)| - |w - (1 + \beta)| > 0$ which implies $Re(w) > \alpha$

Hence the inequality $\begin{array}{r}Re\left\{\frac{(z + \beta z^2)(D_{q,z}^{m + \xi + 1}f(z)) + \lambda(z^2 + \beta z)(D_{q,z}^{m + \xi + 2}f(z))}{(1 - \lambda)(D_{q,z}^{m + \xi}f(z)) + \lambda(z + \beta z^2)(D_{q,z}^{m + \xi + 1}f(z))}\right\} >\alpha \end{array}$

which implies $f(z)\in \Psi_{m,n,p}(\alpha,\beta,\lambda,\xi,\mathbf{q})$

so, the proof of theorem 1 is completed.

Corollary 1: Let the function $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ is a member of new subclass $\Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathbf{q})$ of multivalent function then

where $\mathbf{k} = \mathbf{n} + \mathbf{p}$ , $\mathbf{p}$ is some natural number, $\mathbf{n}$ is a natural number.

In this part of the paper, we derive some results related to Radii of starlikeness, convexity and close to convexity for the function $f(z)$ belonging to the new subclass $\Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$

Theorem 2: Let the function $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and $f(z)$ belong to $\Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathbf{q})$ then the function $f(z)$ is p-valent close to convex of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_{1}^{*}$ , where

Proof: Let $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ and $f(z) = z^p - \sum_{k=n+p}^{\infty} a_k z^k$ .

To prove $f(z)$ is p-valent close to convex of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_1^*$ for this we have to show that

The inequality (3.2) is less than or equal to $p - \varphi$ if

we know that $f(z)\in \Psi_{m,n,p}(\alpha,\beta,\lambda,\xi,\mathfrak{q})$ if and only if

The inequality (3.2) is hold true if

\begin{array}{l}| z | < r_{1}^* \\ = \inf_{k \geq n + p} \{\left(\frac{p - \varphi}{k}\right) \left\{\frac{(1 + \beta) [ k - (m + \xi) ]_{q}[ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ]_{q}- \alpha (1 - \lambda) ]}{(1 + \beta) [ p - (m + \xi) ]_{q}[ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ]_{q}- \alpha (1 - \lambda) ]}\right\} E_{p, k} ^ {m, \xi} ^ {\frac{1}{k - p}}\end{array}

Hence, the given function $f(z)$ is p-valent close to convex of order $\varphi$

Theorem 3: Let the function $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathbf{q})$ then the function $f(z)$ is a p-valent starlike of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_2^*$ , where

Proof: Let $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ and $f(z) = z^p - \sum_{k=n+p}^{\infty} a_k z^k$ .

To prove the function $f(z)$ is p-valent starlike of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_2^*$ for this we have to show that

Now we take the L.H.S. part of the inequality (3.7)

The inequality (3.7) is less than or equal to $p - \varphi$ if

The inequality (3.9) is hold true if

Notes or, we have

so we get the required result

\begin{array}{l} 
| z | < r _ {2} \\
= \inf _ {k \geq n + p} \left\{\left(\frac{p - \varphi}{k - \varphi}\right) \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} \left[ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} \right] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} \left[ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} \right] - \alpha (1 - \lambda)} E _ {p, k} ^ {m, \xi} ^ {\frac{1}{k - p}}\right\}\right\}
\end{array}

Hence, the given function $f(z)$ is p-valent starlike of order $\varphi$

Theorem 4: Let the function $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathrm{q})$ then the given function $f(z)$ is a p-valent convex function of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_3^*$ , where

Proof: Let $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ and $f(z) = z^p - \sum_{k=n+p}^{\infty} a_k z^k$ .

To prove the function $f(z)$ is p-valent convex function of order $\varphi$ ; $0 \leq \varphi < p$ in $|z| < r_3^*$ for this we have to show that

Taking the L.H.S. part of the inequality (3.12)

The inequality (3.12) is less than or equal to $p - \varphi$ if

we know that $f(z)\in \Psi_{m,n,p}(\alpha,\beta,\lambda,\xi,\mathbf{q})$ if and only if

The inequality (3.12) is hold true if

or, we have

so we get the required result

Hence, the given function $f(z)$ is p-valent convex function of order $\varphi$