A note on boundedness properties of Wright's generalized hypergeometric functions

Theorem 2. Let $\Delta \neq 0$ , and $$\Delta_{p}\Psi_{q}\left[\begin{smallmatrix} (\lambda_{i}, & A_{i})_{1,p} & ; \\ (\mu_{i}, & B_{i})_{1,q} & ; \end{smallmatrix}\right] \in P(A,B) \text{ and } |z| \leq |B|, \text{ then}$$ $$\left| {}_{\mathcal{D}}\Psi_{q} \left[ \begin{matrix} (\lambda_{i} + A_{i}, A_{i})_{1,p} ; \\ (\mu_{i} + B_{i}, B_{i})_{1,q} ; \end{matrix} z \right] \right| \leq \frac{A - B}{\left| \Delta \right| (1 - |B| |z|^{2})} , \tag{3.1}$$ where $\Delta$ is given by (2.1). Proof. Since $$\Delta p \Psi_q \begin{bmatrix} (\lambda_i, A_i)_{1,p}; \\ (\mu_i, B_i)_{1,q}; \end{bmatrix} z$$ $$= 1 + \Delta \sum_{n=1}^{\infty} \left( \prod_{i=1}^{p} \Gamma \left( \lambda_{i} + A_{i} n \right) \right) \left( \prod_{i=1}^{q} \Gamma \left( \mu_{i} + B_{i} n \right) \right)^{-1} \frac{z^{n}}{n!} ,$$ it follows then that $$\Delta \frac{d}{dz} p \Psi_q \begin{bmatrix} (\lambda_i, A_i)_{1,p}; \\ (\mu_i, B_i)_{1,q}; \end{bmatrix} = \Delta p \Psi_q \begin{bmatrix} (\lambda_i + A_i, A_i)_{1,p}; \\ (\mu_i + B_i, B_i)_{1,q}; \end{bmatrix} . \tag{3.2}$$ We recall the following result ([8]): If $$h(z) = 1 + \sum_{k=1}^{\infty} C_k z^k \in P(A,B), |z| < 1$$ , then $$| h'(z) | \leq \begin{cases} \frac{A - B}{(1 - |B| |z|)^2} & |z| \leq |B|, \\ \frac{A - B}{(1 - |B|^2) (1 - |z|^2)} & |z| > |B|. \end{cases}$$ (3.3) If we set $$h(z) = \Delta p \Psi_q \begin{bmatrix} (\lambda_i, A_i)_{1,p}; \\ (\mu_i, B_i)_{1,q}; \end{bmatrix} \text{ in (3.3),}$$ and use (3.2) in the process, we are lead to the result (3.1).

Theorem 3. Let $\Delta \neq 0$ , and $$\Delta z_{p}\Psi_{q}\left[ (\lambda_{i}, A_{i})_{1,p}; \atop (\mu_{i}, B_{i})_{1,q}; z \right] \in R (A,B),$$ then for $Re(\gamma) > 0$ and |z| < 1: $$\left| p+2\Psi_{q+2} \left[ \begin{array}{cccc} (\lambda_{i}, A_{i})_{1,p}, (2,1), (\gamma+1,1) & ; \\ (\mu_{i}, B_{i})_{1,q}, (1,1), (\gamma+\alpha+1,1) & ; \end{array} \right] \right|$$ $$\leq \left[\Delta \Gamma \left(\alpha + 1\right) \begin{pmatrix} \alpha + \gamma \\ \gamma \end{pmatrix}\right]^{-1} \left(\frac{1 + A |z|}{1 + B |z|}\right), \tag{3.4}$$ and $$Re\left\{ p+2\Psi_{q+2} \left[ \begin{array}{cc} (\lambda_{i}, A_{i})_{1,p}, (2,1), (\gamma+1,1); \\ (\mu_{i}, B_{i})_{1,q}, (1,1), (\gamma+\alpha+1,1); \end{array} z \right] \right\}$$ $$\geq \left[ \Delta \Gamma \left( \alpha + 1 \right) \begin{pmatrix} \alpha + \gamma \\ \gamma \end{pmatrix} \right]^{-1} \left( \frac{1 - A |z|}{1 - B |z|} \right) . \tag{3.5}$$ Proof. Using (1.1) and (1.4), we find that $$\phi_{\gamma}^{\alpha} \left\{ \Delta z_{p} \Psi_{q} \left[ \begin{pmatrix} \lambda_{i}, A_{i} \end{pmatrix}_{1,p}; \\ (\mu_{i}, B_{i})_{1,q}; z \right] \right\}$$ $$= z + \Delta \Gamma(\alpha+1) {\alpha+\gamma \choose \gamma} \sum_{n=1}^{\infty} \frac{\prod_{i=1}^{p} \Gamma(\lambda_i + A_i n) \Gamma(\gamma+1+n)}{\prod_{i=1}^{q} \Gamma(\mu_i + B_i n) \Gamma(\gamma+\alpha+1+n)} \frac{z^{n+1}}{n!}.$$ (3.6) Then $$\frac{d}{dz} \left\{ \varphi_{\gamma}^{\alpha} \left( \Delta z_{p} \Psi_{q} \left[ \begin{pmatrix} \lambda_{i}, A_{i} \end{pmatrix}_{1,p}; \\ (\mu_{i}, B_{i})_{1,q}; z \right] \right) \right\}$$ $$= \Delta\Gamma(\alpha+1) {\alpha+\gamma \choose \gamma} {\Psi_{q+2}} {\{(\lambda_i, A_i)_{1,p}, (2,1), (\gamma+1,1); z\} \atop (\mu_i, B_i)_{1,q}, (1,1), (\gamma+\alpha+1,1); z\}}.$$ $$\text{Taking } G(z) = \phi_{\gamma}^{\alpha} \left( \Delta z {}_{p} \Psi_{q} {\{(\lambda_i, A_i)_{1,p}; z\} \atop (\mu_i, B_i)_{1,q}; z\}} \right)$$ (3.7) in (2.8) and (2.9), and using (3.7), we obtain the inequalities (3.4) and (3.5), respectively.

Corollary 1. L~r Fd + v) jy (z) c P\*(a), ~/: (i) $$\left|\frac{1}{\Gamma(1+\nu+\mu n)}\right| \leq \frac{2(1-\alpha)n!}{\left|\Gamma(1+\nu)\right|},$$ (4.4) (ii) $$|J_{\nu}^{\mu}(z)| \leq \frac{1}{|\Gamma(1+\nu)|} \left( \frac{1+(1-2\alpha)|z|}{1-|z|} \right)$$ , (4.5) (iii) $$\operatorname{Re}\left\{J_{\nu}^{\mu}\left(z\right)\right\} \geq \frac{1}{\Gamma\left(1+\nu\right)}\left(\frac{1-\left(1-2\alpha\right)\left|z\right|}{1+\left|z\right|}\right),$$ (4.6) (iv) $$\left| {}_{1}\Psi_{2} \right|^{(1, 1)} ; z \left| {}_{(1+\nu, \mu), (2, 1);} \right|$$ $$\geq \frac{(2\alpha-1)}{|\Gamma(1+\nu)|} \left[ 1 + \frac{2(1-\alpha)}{1-2\alpha} \cdot \frac{\log(1-|z|)}{|z|} \right], \tag{4.7}$$ (v) $$\left| \Psi_{2} \begin{bmatrix} (1, 1) \\ (1+\nu, \mu), (2, 1) \end{bmatrix} \right|$$ $$\geq \frac{(2\alpha-1)}{|\Gamma(1+\nu)|} \left[ 1 - \frac{2(1-\alpha)}{1-2\alpha} \cdot \frac{\log(1+|z|)}{|z|} \right], \tag{4.8}$$

Corollary 2. If r i + v >Jf z >e P \* a > ,and ) z ] S I ,then $$\left| J_{1+\nu+\mu}^{\mu} \left( z \right) \right| \leq \frac{2\left( 1-\alpha \right)}{\left| \Gamma\left( 1+\nu \right) \right| \left( 1+\left| z \right|^{2} \right)} .$$ (4.9)

Corollary 3. Let r (03B2) > z i " # > /« E03B1, 03B2 z ) e P\* a ), , then (i) $$\left|\frac{1}{\Gamma(\beta + \alpha n)}\right| \le \frac{2(1-\alpha)n!}{|\Gamma(\beta)|}$$ , (4.10) (ii) $$|z^{(1-\beta)/\alpha} E_{\alpha,\beta}(z)| \le \frac{1}{|\Gamma(\beta)|} \left( \frac{1 + (1-2\alpha)|z|}{1 + |z|} \right)$$ , (4.11) (iii) $$Re\{z^{(1-\beta)/\alpha} E_{\alpha,\beta}(z)\} \ge \frac{1}{|\Gamma(\beta)|} \left(\frac{1-(1-2\alpha)|z|}{1+|z|}\right),$$ (4.12) (iv) $$\left| {}_{2}\Psi_{2} \left[ {}_{(\beta,\alpha)}^{(1,1)}, {}_{(2,1)}^{(1,1)}; z \right] \right|$$ $$\leq \frac{(2 \alpha - 1)}{|\Gamma(\beta)|} \left[ 1 + \frac{2(1 - \alpha)}{1 - 2 \alpha} \cdot \frac{\log(1 - |z|)}{|z|} \right] \tag{4.13}$$ (v) $$\left[ 2\Psi_{2} \left[ \begin{pmatrix} 1,1 \\ \beta,\alpha \end{pmatrix}, \begin{pmatrix} 1,1 \\ 2,1 \end{pmatrix}; z \right] \right]$$ $$\geq \frac{(2\alpha - 1)}{|\Gamma(\beta)|} \left[ 1 - \frac{2(1-\alpha)}{1-2\alpha} \cdot \frac{\log(1+|z|)}{|z|} \right], \tag{4.14}$$

Corollary 4. If 0393 (03B2) z(1-03B2)/03B1 E03B1,03B2 (z> e P\* «) , and ] z ] I, then $$\left| \mathbf{1} \Psi_{1} \left[ \begin{pmatrix} (2,1) \\ (\beta+\alpha,\alpha) \end{pmatrix}^{\prime}, z \right] \right| \leq \frac{2 (1-\alpha)}{\left| \Gamma(\beta) \right| (1+|z|^{2})}.$$