## 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 $$ f (z) = z ^ {p} - \sum_ {k = n + p} ^ {\infty} a _ {k} z ^ {k} \tag {1.1} $$ 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 $$ U _ {1} = \{z: | z | < 1 \} $$ If a function $f(z) \in \mathcal{H}(p)$ satisfies the following condition $$ R e \left\{\frac {z f ^ {'} (z)}{f (z)} \right\} > \varphi z \in U _ {1}, 0 \leq \varphi < p, p \in \mathbb {N} \tag {1.2} $$ 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 $$ R e \left\{1 + \frac {z f ^ {\prime \prime} (z)}{f ^ {\prime} (z)} \right\} > \varphi z \in U _ {1}, 0 \leq \varphi < p, p \in \mathbb {N} \tag {1.3} $$ 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 $$ (f * g) (z) = (g * f) (z) = z ^ {p} - \sum_ {k = n + p} ^ {\infty} a _ {k} b _ {k} z ^ {k} \tag {1.4} $$ 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 $$ D _ {q, z} f (z) = \frac{f (z) - f (z q)}{z - z q} z \neq 0 and q \neq 1 \tag{1.5} $$ The Jackson's q - derivative tends to ordinary derivative when q tends to 1. The Jackson q- derivative can also be written as $$ D _ {q, z} ^ {\mathrm{m}} z ^ {r} = \frac{\Gamma_ {q} (1 + r)}{\Gamma_ {q} (1 + r - m)} z ^ {r - m} \quad \text{where} m \geq 0, r > - 1 \tag{1.6} $$ 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 $$ R e \left\{\frac {\left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right) + \lambda \left(z ^ {2} + \beta z\right) \left(D _ {q , z} ^ {m + \xi + 2} f (z)\right)}{(1 - \lambda) \left(D _ {q , z} ^ {m + \xi} f (z)\right) + \lambda \left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right)} \right\} > \alpha \tag {1.7} $$ 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: 1. If $m = 0, \beta = 0, \xi = 0, q \to 1$ then from (1.7) we get $$ R e \left\{\frac {z (D _ {1 , z} f (z)) + \lambda z ^ {2} (D _ {1 , z} ^ {2} f (z))}{(1 - \lambda) (D _ {1 , z} ^ {0} f (z)) + \lambda z (D _ {1 , z} f (z))} \right\} > \alpha $$ which is equivalent to $$ R e \left\{\frac {z f ^ {\prime} (z) + \lambda z ^ {2} f ^ {\prime \prime} (z)}{(1 - \lambda) f (z) + \lambda z f ^ {\prime} (z)} \right\} > \alpha $$ 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] 2. If $m = 0, \beta = 0, q \to 1$ then from (1.7) we get $$ R e \left\{\frac {z (D _ {1 , z} ^ {\xi + 1} f (z)) + \lambda z ^ {2} (D _ {1 , z} ^ {\xi + 2} f (z))}{(1 - \lambda) (D _ {1 , z} ^ {\xi} f (z)) + \lambda z (D _ {1 , z} ^ {\xi + 1} f (z))} \right\} > \alpha $$ 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] 3. If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0$ then from (1.7) we get $$ R e \left\{\frac {z f ^ {'} (z)}{f (z)} \right\} > \alpha \qquad 0 \leq \alpha < p $$ 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 $\alpha$. 4. If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0, p = 1$ then from (1.7) we get $$ R e \left\{\frac {z f ^ {'} (z)}{f (z)} \right\} > \alpha \qquad 0 \leq \alpha < 1 $$ so $\Psi_{0,n,1}(\alpha, 0,0,0,1) \equiv \mathrm{T}^{*}(1, \alpha)$, which was earlier studied by Srivastava et al. [15]. 5. If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 0, p = 1, n = 1$ then from (1.7) we get $$ R e \left\{\frac {z f ^ {'} (z)}{f (z)} \right\} > \alpha \qquad 0 \leq \alpha < 1 $$ Then we get a class which was earlier discussed by Silverman [14]. 6. If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 1$ then from (1.7) we get $$ R e \left\{\frac {z f ^ {\prime} (z) + z ^ {2} f ^ {\prime \prime} (z)}{z f ^ {\prime} (z)} \right\} > \alpha \quad 0 \leq \alpha < p $$ 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 $\alpha$. 7. If $m = 0, \beta = 0, \xi = 0, q \to 1, \lambda = 1, p = 1$ then from (1.7) we get $$ R e \left\{1 + \frac {z f ^ {\prime \prime} (z)}{f ^ {\prime} (z)} \right\} > \alpha \qquad 0 \leq \alpha < 1 $$ 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 $$ R e \left\{1 + \frac {z f ^ {\prime \prime} (z)}{f ^ {\prime} (z)} \right\} > \alpha \quad 0 \leq \alpha < 1 $$ 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 $$ \sum_{k = n + p}^\infty E_{p,k}^{m,\xi} \left\{ \frac{(1 + \beta) [k - (m + \xi)]_q [1 - \alpha\lambda + \lambda [k - (m + \xi + 1)]_q] - \alpha(1 - \lambda)}{(1 + \beta) [p - (m + \xi)]_q [1 - \alpha\lambda + \lambda [p - (m + \xi + 1)]_q] - \alpha(1 - \lambda)} \right\} a_k \leq 1 $$ $$ where E_{p,k}^{m,\xi} = \frac{\Gamma_q(1+k)\Gamma_q(1+p-(m+\xi))}{\Gamma_q(1+p)\Gamma_q(1+k-(m+\xi))} $$ $$ z \in U _ {1}, m \in \mathbb{N} \cup \{0 \}, 0 \leq \alpha < p, \qquad 0 \leq \beta < 1, 0 \leq \lambda \leq 1 and 0 \leq \xi < 1 $$ Proof: Let us consider that $f(z) \in \Psi_{m,n,p}(\alpha, \beta, \lambda, \xi, \mathfrak{q})$ so we have $$ R e \left\{\frac {\left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right) + \lambda \left(z ^ {2} + \beta z\right) \left(D _ {q , z} ^ {m + \xi + 2} f (z)\right)}{(1 - \lambda) \left(D _ {q , z} ^ {m + \xi} f (z)\right) + \lambda \left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right)} \right\} > \alpha $$ Since $f(z) = z^{p} - \sum_{k=n+p}^{\infty} a_{k}z^{k}$ and $$ D_{q,z}^{m+\xi} f(z) = \frac{\Gamma_q(1+p)}{\Gamma_q(1+p-(m+\xi))} z^{p-(m+\xi)} - \sum_{k=n+p}^{\infty} \frac{\Gamma_q(1+k)}{\Gamma_q(1+k-(m+\xi))} a_k z^{k-(m+\xi)} \quad (2.2) $$ so we have $$ D_{q,z}^{m+\xi+1} = \frac{\Gamma_q(1+p)}{\Gamma_q(p-(m+\xi))} z^{p-(m+\xi+1)} - \sum_{k=n+p}^{\infty} \frac{\Gamma_q(1+k)}{\Gamma_q(k-(m+\xi))} a_k z^{k-(m+\xi+1)} \qquad (2.3) $$ $$ D _ {q, z} ^ {m + \xi + 2} = \frac{\Gamma_ {q} (1 + p)}{\Gamma_ {q} (p - (m + \xi + 1))} z ^ {p - (m + \xi + 2)} - \sum_ {k = n + p} ^ {\infty} \frac{\Gamma_ {q} (1 + k)}{\Gamma_ {q} (k - (m + \xi + 1))} a _ {k} z ^ {k - (m + \xi + 2)} \qquad (2.4) $$ 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 $\mathbf{D}$ $$ \mathrm{N} = (z + \beta z^{2}) \left[ \frac{\Gamma_{q}(1+p)}{\Gamma_{q}(p-(m+\xi))} z^{p-(m+\xi+1)} - \sum_{k = n+p}^{\infty} \frac{\Gamma_{q}(1+k)}{\Gamma_{q}(k-(m+\xi))} a_{k} z^{k-(m+\xi+1)} \right] + $$ $$ \lambda (z ^ {2} + \beta z) \left[ \frac {\Gamma_ {q} (1 + p)}{\Gamma_ {q} (p - (m + \xi + 1))} z ^ {p - (m + \xi + 2)} - \sum_ {k = n + p} ^ {\infty} \frac {\Gamma_ {q} (1 + k)}{\Gamma_ {q} (k - (m + \xi + 1))} a _ {k} z ^ {k - (m + \xi + 2)} \right] $$ $$ \mathrm{D} = (1 - \lambda) \left[ \frac{\Gamma_ {q} (1 + p)}{\Gamma_ {q} (1 + p - (m + \xi))} z ^ {p - (m + \xi)} - \sum_ {k = n + p} ^ {\infty} \frac{\Gamma_ {q} (1 + k)}{\Gamma_ {q} (1 + k - (m + \xi))} a _ {k} z ^ {k - (m + \xi)} \right] + $$ $$ \lambda \left(z + \beta z ^ {2}\right) \left[ \frac {\Gamma_ {q} \left(1 + p\right)}{\Gamma_ {q} \left(p - (m + \xi)\right)} z ^ {p - (m + \xi + 1)} - \sum_ {k = n + p} ^ {\infty} \frac {\Gamma_ {q} \left(1 + k\right)}{\Gamma_ {q} \left(k - (m + \xi)\right)} a _ {k} z ^ {k - (m + \xi + 1)} \right] $$ 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 $$ \frac {\Gamma_ {q} (1 + p)}{\Gamma_ {q} \big (1 + p - (m + \xi) \big)} \Big [ (1 + \beta) [ p - (m + \xi) ] _ {q} \big [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} \big ] - \alpha (1 - \lambda) \Big ] $$ $$ \geq \sum_{k=n+p}^\infty \frac{\Gamma_q(1+k)}{\Gamma_q(1+k-(m+\xi))} a_k \big[ (1+\beta)[k-(m+\xi)]_q \big[ 1-\alpha\lambda+\lambda[k-(m+\xi+1)]_q \big] - \alpha(1-\lambda) \big] $$ on simplifying we get, $$ \sum_{k=n+p}^{\infty} E_{p,k}^{m,\xi} \left\{ \frac{(1+\beta)[k-(m+\xi)]_q[1-\alpha\lambda+\lambda[k-(m+\xi+1)]_q]-\alpha(1-\lambda)}{(1+\beta)[p-(m+\xi)]_q[1-\alpha\lambda+\lambda[p-(m+\xi+1)]_q]-\alpha(1-\lambda)} \right\} a_k \leq 1 $$ $$ \mathrm {w h e r e} E _ {p, k} ^ {m, \xi} = \frac {\Gamma_ {q} (1 + k) \Gamma_ {q} (1 + p - (m + \xi))}{\Gamma_ {q} (1 + p) \Gamma_ {q} (1 + k - (m + \xi))} $$ 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 $$ Re\left\{\frac{(z + \beta z^{2})\left(D_{q,z}^{m+\xi+1}f(z)\right) + \lambda(z^{2} + \beta z)\left(D_{q,z}^{m+\xi+2}f(z)\right)}{(1 - \lambda)\left(D_{q,z}^{m+\xi}f(z)\right) + \lambda(z + \beta z^{2})\left(D_{q,z}^{m+\xi+1}f(z)\right)}\right\} > \alpha $$ 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) $$ L = \left| \frac{(z + \beta z^2) (D_{q,z}^{m + \xi + 1} f(z)) + \lambda (z^2 + \beta z) (D_{q,z}^{m + \xi + 2} f(z))}{(1 - \lambda) (D_{q,z}^{m + \xi} f(z)) + \lambda (z + \beta z^2) (D_{q,z}^{m + \xi + 1} f(z))} - (1 + \alpha) \right| \tag{2.7} $$ and$K = |w + (1 - \alpha)|$. $$ K = \left| \frac {\left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right) + \lambda \left(z ^ {2} + \beta z\right) \left(D _ {q , z} ^ {m + \xi + 2} f (z)\right)}{(1 - \lambda) \left(D _ {q , z} ^ {m + \xi} f (z)\right) + \lambda \left(z + \beta z ^ {2}\right) \left(D _ {q , z} ^ {m + \xi + 1} f (z)\right)} + (1 - \alpha) \right| \tag {2.8} $$ 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 $$ a_{k} \leq \left\{\frac{(1 + \beta) [ p - (m + \xi) ]_{q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ]_{q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ k - (m + \xi) ]_{q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ]_{q} ] - \alpha (1 - \lambda)} \right\} \frac{1}{E_{p,k}^{m,\xi}} \tag{2.9} $$ where $\mathbf{k} = \mathbf{n} + \mathbf{p}$, $\mathbf{p}$ is some natural number, $\mathbf{n}$ is a natural number. # III. PROPERTY OF NEW SUBCLASS RELATED TO RADI OF STAR LIKENESS, CONVEXITY AND 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 $$ r_{1}^{*} = \inf_{k \geq n + p} \left\{ \left( \frac{p - \varphi}{k} \right) \left\{ \frac{(1 + \beta) [k - (m + \xi)]_{q} [1 - \alpha \lambda + \lambda [k - (m + \xi + 1)]_{q}] - \alpha (1 - \lambda)}{(1 + \beta) [p - (m + \xi)]_{q} [1 - \alpha \lambda + \lambda [p - (m + \xi + 1)]_{q}] - \alpha (1 - \lambda)} \right\} E_{p, k}^{m, \xi} \right\}^{\frac{1}{k - p}} \tag{3.1} $$ 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 $$ \left| \frac{f ^ {'} (z)}{z ^ {p - 1}} - p \right| \leq p - \varphi \quad | z | < r _ {1} * \tag{3.2} $$ $$ \begin{array}{l} \left| \frac{f ^ {\prime} (z)}{z ^ {p - 1}} - p \right| = \left| \frac{p z ^ {p - 1} - \sum_ {k = n + p} ^ {\infty} k a _ {k} z ^ {k - 1}}{z ^ {p - 1}} - p \right| \\= \left| \frac{\sum_ {k = n + p} ^ {\infty} k a _ {k} z ^ {k - 1}}{z ^ {p - 1}} \right| \\\leq \sum_ {k = n + p} ^ {\infty} k a _ {k} | z | ^ {k - p} \tag{3.3} \\\end{array} $$ The inequality (3.2) is less than or equal to $p - \varphi$ if $$ \sum_ {k = n + p} ^ {\infty} \left(\frac{k}{p - \varphi}\right) a _ {k} | z | ^ {k - p} \leq 1 \tag{3.4} $$ we know that $f(z)\in \Psi_{m,n,p}(\alpha,\beta,\lambda,\xi,\mathfrak{q})$ if and only if $$ \sum_ {k = n + p} ^ {\infty} E _ {p, k} ^ {m, \xi} \left\{\frac {(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)} \right\} a _ {k} \leq 1 $$ The inequality (3.2) is hold true if $$ \begin{array}{l} \left(\frac{k}{p - \varphi}\right) | z | ^ {k - p} \\\leq E _ {p, k} ^ {m, \xi} \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}\right\} \\\end{array} $$ $$ \left| z \right| ^ {k - p} \leq \left(\frac{p - \varphi}{k}\right) E _ {p, k} ^ {m, \xi} \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}\right\} \tag{3.5} $$ $$ \begin{array}{l} | z | < r_{1}^* \\ = \inf_{k \geq n + p} \left\{\left(\frac{p - \varphi}{k}\right) \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} - \alpha (1 - \lambda) ]}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} - \alpha (1 - \lambda) ]}\right\} E_{p, k} ^ {m, \xi} ^ {\frac{1}{k - p}} \end{array} $$ 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 $$ r_{2}^* = \inf_{k \geq n + p} \left\{ \left( \frac{p - \varphi}{k - \varphi} \right) \left\{ \frac{(1 + \beta) [k - (m + \xi)]_q [1 - \alpha \Lambda + \Lambda [k - (m + \xi + 1)]_q] - \alpha (1 - \Lambda)}{(1 + \beta) [p - (m + \xi)]_q [1 - \alpha \Lambda + \Lambda [p - (m + \xi + 1)]_q] - \alpha (1 - \Lambda)} \right\} E_{p,k}^{m,\xi} \right\}^{\frac{1}{k - p}} \tag{3.6} $$ 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 $$ \left| \frac{z f ^ {'} (z)}{f (z)} - p \right| \leq p - \varphi \quad | z | < r _ {2} * \tag{3.7} $$ Now we take the L.H.S. part of the inequality (3.7) $$ \begin{array}{l} \left| \frac{z f ^ {\prime} (z)}{f (z)} - p \right| = \left| \frac{z (p z ^ {p - 1} - \sum_ {k = n + p} ^ {\infty} k a _ {k} z ^ {k - 1})}{z ^ {p} - \sum_ {k = n + p} ^ {\infty} a _ {k} z ^ {k}} - p \right| \\= \left| \frac{\sum_ {k = n + p} ^ {\infty} (k - p) a _ {k} z ^ {k}}{z ^ {p} - \sum_ {k = n + p} ^ {\infty} a _ {k} z ^ {k}} \right| \\\leq \frac{\sum_ {k = n + p} ^ {\infty} (k - p) a _ {k} | z | ^ {k - p}}{1 - \sum_ {k = n + p} ^ {\infty} a _ {k} | z | ^ {k - p}} \tag{3.8} \\\end{array} $$ The inequality (3.7) is less than or equal to $p - \varphi$ if $$ \sum_ {k = n + p} ^ {\infty} \frac{(k - \varphi)}{(p - \varphi)} a _ {k} | z | ^ {k - p} \leq 1 \tag{3.9} $$ $$ \sum_{k=n+p}^\infty E_{p,k}^{m,\xi} \left\{\frac{(1+\beta)[k-(m+\xi)]_q[1-\alpha\lambda+\lambda[k-(m+\xi+1)]_q]-\alpha(1-\lambda)}{(1+\beta)[p-(m+\xi)]_q[1-\alpha\lambda+\lambda[p-(m+\xi+1)]_q]-\alpha(1-\lambda)}\right\} a_k \leq 1 $$ The inequality (3.9) is hold true if $$ \begin{array}{l} \left(\frac{k - \varphi}{p - \varphi}\right) | z | ^ {k - p} \\\leq E _ {p, k} ^ {m, \xi} \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}\right\} \\\end{array} $$ Notes or, we have $$ | z | ^ {k - p} \leq \left(\frac{p - \varphi}{k - \varphi}\right) E _ {p, k} ^ {m, \xi} \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}\right\} \tag{3.10} $$ 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 $$ r_{3}^* = \inf_{k \geq n + p} \left\{ \frac{p}{k} \left( \frac{p - \varphi}{k - \varphi} \right) \left\{ \frac{(1 + \beta) [k - (m + \xi)]_q [1 - \alpha \lambda + \lambda [k - (m + \xi + 1)]_q] - \alpha (1 - \lambda)}{(1 + \beta) [p - (m + \xi)]_q [1 - \alpha \lambda + \lambda [p - (m + \xi + 1)]_q] - \alpha (1 - \lambda)} \right\} E_{p,k}^{m,\xi} \right\}^{\frac{1}{k - p}} \tag{3.11} $$ 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 $$ \left| \frac{z f ^ {\prime \prime} (z)}{f ^ {\prime} (z)} + (1 - p) \right| \leq p - \varphi \quad | z | < r _ {3} * \tag{3.12} $$ Taking the L.H.S. part of the inequality (3.12) $$ \left| \frac{z f ^ {\prime \prime} (z)}{f ^ {\prime} (z)} + (1 - p) \right| = \left| \frac{z (p (p - 1) z ^ {p - 2} - \sum_ {k = n + p} ^ {\infty} k (k - 1) a _ {k} z ^ {k - 2})}{p z ^ {p - 1} - \sum_ {k = n + p} ^ {\infty} k a _ {k} z ^ {k - 1}} + (1 - p) \right| \\= \left| \frac{\sum_ {k = n + p} ^ {\infty} k (k - p) a _ {k} z ^ {k - p}}{p - \sum_ {k = n + p} ^ {\infty} k \alpha_ {k} z ^ {k - p}} \right| $$ $$ \leq \frac{\sum_ {k = n + p} ^ {\infty} k (k - p) a _ {k} | z | ^ {k - p}}{p - \sum_ {k = n + p} ^ {\infty} k a _ {k} | z | ^ {k - p}} $$ The inequality (3.12) is less than or equal to $p - \varphi$ if $$ \sum_ {k = n + p} ^ {\infty} \frac{k (k - \lambda)}{p (p - \lambda)} a _ {k} | z | ^ {k - p} \leq 1 \tag{3.13} $$ we know that $f(z)\in \Psi_{m,n,p}(\alpha,\beta,\lambda,\xi,\mathbf{q})$ if and only if $$ \sum_{k=n+p}^\infty E_{p,k}^{m,\xi} \left\{ \frac{(1+\beta)[k-(m+\xi)]_q[1-\alpha\lambda+\lambda[k-(m+\xi+1)]_q]-\alpha(1-\lambda)}{(1+\beta)[p-(m+\xi)]_q[1-\alpha\lambda+\lambda[p-(m+\xi+1)]_q]-\alpha(1-\lambda)} \right\} a_k \leq 1 $$ The inequality (3.12) is hold true if $$ \frac{k}{p} \left(\frac{k - \varphi}{p - \varphi}\right) |z|^{k - p} \leq E_{p,k}^{m,\xi} \left\{\frac{(1 + \beta) [k - (m + \xi)]_q [1 - \alpha\lambda + \lambda [k - (m + \xi + 1)]_q] - \alpha(1 - \lambda)}{(1 + \beta) [p - (m + \xi)]_q [1 - \alpha\lambda + \lambda [p - (m + \xi + 1)]_q] - \alpha(1 - \lambda)}\right\} $$ or, we have $$ \left| z \right| ^ {k - p} \leq \frac{p}{k} \left(\frac{p - \varphi}{k - \varphi}\right) E _ {p, k} ^ {m, \xi} \left\{\frac{(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)} \right\} \tag{3.14} $$ so we get the required result $$ \begin{array}{l} | z | < r _ {3} ^ {*} \\= \inf _ {k \geq n + p} \left\{\frac {p}{k} \Big (\frac {p - \varphi}{k - \varphi} \Big) \Bigg \{\frac {(1 + \beta) [ k - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ k - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)}{(1 + \beta) [ p - (m + \xi) ] _ {q} [ 1 - \alpha \lambda + \lambda [ p - (m + \xi + 1) ] _ {q} ] - \alpha (1 - \lambda)} \Bigg \} E _ {p, k} ^ {m, \xi} \right\} ^ {\frac {1}{k - p}} \\\end{array} $$ Hence, the given function $f(z)$ is p-valent convex function of order $\varphi$

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