Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions

Theorem 2.1. Let the function f(z) be defined by (1.1). If (2.1) $$\sum_{n=1}^{\infty} \frac{(2+\alpha)_n}{(1+\alpha)_n} \left| \frac{(a)_n}{(c)_n} \right| |a_{n+1}| \le 1$$ for $\alpha > -1$ . then $f(z) \in \mathscr{V}(a, c; \alpha)$ . The result (2.1) is sharp.

Corollary 2.1. Let the function f(z) be defined by (1.1). If (2.8) $$\sum_{n=1}^{\infty} \frac{(2+\alpha)_n}{(1+\alpha)_n} |a_{n+1}| \le 1$$ for $\alpha > -1$ , then $f(z) \in \mathcal{V}(\alpha)$ . The result (2.8) is sharp for the functions given by (2.9) $$f(z) = z + \frac{(1+\alpha)_n}{(2+\alpha)_n} z^{n+1} \qquad (n \in \mathcal{N}).$$ Remark 2.1. In its special case when $-1 < \alpha \le 0$ , Corollary 2.1 yields the corresponding result for the class $\mathscr{S}^*(\alpha)$ due to Silverman [14, p. 110, Theorem 1].

Theorem 2.2. Let the function f(z) be defined by (1.1). If (2.10) $$\sum_{n=1}^{\infty} \frac{(2)_n (2+\alpha)_n}{(1)_n (1+\alpha)_n} \left| \frac{(a)_n}{(c)_n} \right| |a_{n+1}| \leq 1$$ for $\alpha > -1$ , then $f(z) \in \mathcal{W}(a, c; \alpha)$ . The result (2.10) is sharp.

Corollary 2.2. Let the function f(z) be defined by (1.1). If (2.12) $$\sum_{n=1}^{\infty} \frac{(2)_n (2+\alpha)_n}{(1)_n (1+\alpha)_n} |a_{n+1}| \leq 1$$ for $\alpha > -1$ , then $f(z) \in \mathcal{W}(\alpha)$ . The result (2.12) is sharp for the functions given by (2.13) $$f(z) = z + \frac{(1)_n (1 + \alpha)_n}{(2)_n (2 + \alpha)_n} z^{n+1} \qquad (n \in \mathcal{N}).$$ Remark 2.2. If, in Corollary 2.2, we assume that $-1 < \alpha \le 0$ , we shall obtain the corresponding result for the class $\mathcal{K}(\alpha)$ due to Silverman [14, p. 110, Corollary]. For the classes $\mathscr{V}_0(\alpha, c; \alpha)$ and $\mathscr{W}_0(\alpha, c; \alpha)$ , we next prove Theorem 2.3 and Theorem 2.4 below.

Theorem 2.3. Let the function f(z) be defined by (1.26). Then $f(z) \in \mathscr{V}_0(a,c;\alpha)$ if and only if (2.14) $$\sum_{n=1}^{\infty} \frac{(2+\alpha)_n (a)_n}{(1+\alpha)_n (c)_n} |a_{n+1}| \leq 1$$ for $\alpha > -1$ , where $(a)_n/(c)_n > 0$ . The result (2.14) is sharp.

Corollary 2.3. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(a,c;\alpha)$ with $(a)_n/(c)_n > 0$ . Then $$|a_{n+1}| \leq \frac{(1+\alpha)_n(c)_n}{(2+\alpha)_n(a)_n} \qquad (n \in \mathcal{N}).$$ The result (2.18) is sharp for the functions f(z) given by (2.17).

Corollary 2.4. Let the function f(z) be defined by (1.26). Then $f(z) \in \mathscr{V}_0(\alpha)$ if and only if (2.19) $$\sum_{n=1}^{\infty} \frac{(2+\alpha)_n}{(1+\alpha)_n} |a_{n+1}| \leq 1$$ for $\alpha > -1$ . The result (2.19) is sharp for the functions given by (2.20) $$f(z) = z - \frac{(1+\alpha)_n}{(2+\alpha)_n} z^{n+1} (n \in \mathcal{N}).$$ Remark 2.3. For $\alpha$ constrained by $-1 < \alpha \le 0$ , Corollary 2.4 corresponds to a result for the class $\mathcal{F}^*(\alpha)$ proved by Silverman [14, p. 110, Theorem 2].

Theorem 2.4. Let the function f(z) be defined by (1.26). Then $f(z) \in \mathcal{W}_0(a, c; \alpha)$ if and only if (2.21) $$\sum_{n=1}^{\infty} \frac{(2)_n (2+\alpha)_n (\alpha)_n}{(1)_n (1+\alpha)_n (c)_n} |a_{n+1}| \leq 1$$ for $\alpha > -1$ , where $(a)_n/(c)_n > 0$ . The result (2.21) is sharp.

Corollary 2.5. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(a, c; \alpha)$ with $(a)_n/(c)_n > 0$ . Then $$|a_{n+1}| \leq \frac{(1)_n (1+\alpha)_n (c)_n}{(2)_n (2+\alpha)_n (a)_n} \qquad (n \in \mathcal{N}).$$ The result (2.23) is sharp for the functions f(z) given by (2.22).

Corollary 2.6. Let the function f(z) be defined by (1.26). Then $f(z) \in \mathcal{W}_0(\alpha)$ if and only if (2.24) $$\sum_{n=1}^{\infty} \frac{(2)_n (2+\alpha)_n}{(1)_n (1+\alpha)_n} |a_{n+1}| \le 1$$ for $\alpha > -1$ . The result (2.24) is sharp for the functions given by (2.25) $$f(z) = z - \frac{(1)_n (1 + \alpha)_n}{(2)_n (2 + \alpha)_n} z^{n+1} \qquad (n \in \mathcal{N}).$$ Remark 2.4. Corollary 2.6 corresponds, for $-1 < \alpha \le 0$ , to a result for the class $\mathcal{C}(\alpha)$ given by Silverman [14, p. 111, Corollary 2].

Lemma 3.1. The class $\mathscr{V}_{0}(a, c; \alpha)$ with $(a)_{n}/(c)_{n} > 0$ is convex considered as a linear space over the field of real numbers.

Lemma 3.2. Let $$(3.4) f_0(z) = z$$ and (3.5) $$f_n(z) = z - \frac{(1+\alpha)_n(c)_n}{(2+\alpha)_n(a)_n} z^{n+1} (n \in \mathcal{N})$$ with $(a)_n/(c)_n > 0$ . Then $f(z) \in \mathscr{V}_0(a, c; \alpha)$ if and only if it can be expressed in the form $$f(z) = \sum_{n=0}^{\infty} \lambda_n f_n(z) ,$$ where $\lambda_n \geq 0$ , $\forall n \in \mathcal{N} \cup \{0\}$ , and $$\sum_{n=0}^{\infty} \lambda_n = 1.$$ Remark 3.1. Stated simply, Lemma 3.2 asserts that the family $\mathscr{V}_0(a,c;\alpha)$ is an infinite-dimensional simplex with definite vertices defined by certain extremal conditions.

Theorem 3.1. The extreme points of the class $\mathscr{V}_0(a,c;\alpha)$ with $(a)_n/(c)_n > 0$ are the functions $f_n(z)$ $(n \ge 0)$ given by (3.4) and (3.5). In precisely the same manner, we can establish

Theorem 3.2. The extreme points of the class $\mathcal{W}_0(a, c; \alpha)$ with $(a)_n/(c)_n > 0$ are the functions $f_n(z)$ $(n \ge 0)$ given by (3.4) and (3.12) $$f_n(z) = z - \frac{(1)_n (1 + \alpha)_n (c)_n}{(2)_n (2 + \alpha)_n (a)_n} z^{n+1} (n \in \mathcal{N}) .$$ Remark 3.2. As already observed in Remark 3.1, each of Theorem 3.1 and Theorem 3.2 can indeed be restated in terms of infinite-dimensional simplexes. Corresponding to Theorem 3.1 and Theorem 3.2, we now prove the following distortion theorems for functions belonging to the classes $\mathcal{V}_0(a, c; \alpha)$ and $\mathcal{W}_0(a, c; \alpha)$ .

Theorem 3.3. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(a,c;\alpha)$ with $(a)_n/(c)_n>0$ and $a\geq c$ . Then (3.13) $$|z| - \frac{(1+\alpha)c}{(2+\alpha)a}|z|^2 \le |f(z)| \le |z| + \frac{(1+\alpha)c}{(2+\alpha)a}|z|^2$$ for $z \in \mathcal{U}$ . Furthermore, if either $$\alpha \geq 0$$ and $c \leq \min(\frac{1}{2}a, a - 1)$ or $$-1 < \alpha < 0$$ and $c > -1$ . then (3.14) $$1 - \frac{2(1+\alpha)c}{(2+\alpha)a}|z| \le |f'(z)| \le 1 + \frac{2(1+\alpha)c}{(2+\alpha)a}|z|$$ for $z \in \mathcal{U}$ . The results (3.13) and (3.14) are sharp.

Corollary 3.1. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(a,c;\alpha)$ with $(a)_n/(c)_n>0$ and $a\geq c$ . Then the unit disk $\mathscr U$ is mapped onto a domain that contains the disk $|w|< r_0$ , where $r_0$ is given by (3.23) $$r_0 = 1 - \frac{(1+\alpha)c}{(2+\alpha)a}.$$

Corollary 3.2. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(\alpha)$ . Then $$(3.24) |z| - \left(\frac{1+\alpha}{2+\alpha}\right)|z|^2 \le |f(z)| \le |z| + \left(\frac{1+\alpha}{2+\alpha}\right)|z|^2$$ for $z \in \mathcal{U}$ . Furthermore (3.25) $$1 - \frac{2(1+\alpha)}{2+\alpha}|z| \le |f'(z)| \le 1 + \frac{2(1+\alpha)}{2+\alpha}|z|$$ for $z \in \mathcal{U}$ . The results (3.24) and (3.25) are sharp for the function $$(3.26) f(z) = z - \left(\frac{1+\alpha}{2+\alpha}\right)z^2.$$ Remark 3.3. In its special case when $-1 < \alpha \le 0$ , Corollary 3.2 yields the corresponding distortion theorem for the class $\mathcal{F}^*(\alpha)$ proved by Silverman [14, p. 111, Theorem 4]. Applying the above technique mutatis mutandis, we can prove

Theorem 3.4. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(a, c; \alpha)$ with $(a)_n/(c)_n > 0$ and $a \ge c$ . Then $$(3.27) |z| - \frac{(1+\alpha)c}{2(2+\alpha)a}|z|^2 \le |f(z)| \le |z| + \frac{(1+\alpha)c}{2(2+\alpha)a}|z|^2$$ for $z \in \mathcal{U}$ . Furthermore (3.28) $$1 - \frac{(1+\alpha)c}{(2+\alpha)a}|z| \le |f'(z)| \le 1 + \frac{(1+\alpha)c}{(2+\alpha)a}|z|$$ for $z \in \mathcal{U}$ . The results (3.27) and (3.28) are sharp for the function (3.29) $$f(z) = z - \frac{(1+\alpha)c}{2(2+\alpha)a}z^2.$$

Corollary 3.3. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(a,c;\alpha)$ with $(a)_n/(c)_n>0$ and $a\geq c$ . Then the unit disk $\mathcal U$ is mapped onto a domain that contains the disk $|w|< r_1$ , where $r_1$ is given by (3.30) $$r_1 = 1 - \frac{(1+\alpha)c}{2(2+\alpha)a} .$$

Corollary 3.4. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(\alpha)$ . Then (3.31) $$|z| - \frac{1+\alpha}{2(2+\alpha)}|z|^2 \le |f(z)| \le |z| + \frac{1+\alpha}{2(2+\alpha)}|z|^2$$ for $z \in \mathcal{U}$ . Furthermore $$(3.32) 1 - \left(\frac{1+\alpha}{2+\alpha}\right)|z| \le |f'(z)| \le 1 + \left(\frac{1+\alpha}{2+\alpha}\right)|z|$$ for $z \in \mathcal{U}$ . The results (3.31) and (3.32) are sharp for the function (3.33) $$f(z) = z - \frac{1+\alpha}{2(2+\alpha)}z^2.$$

Theorem 4.1. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(a,c;\alpha)$ with $(a)_n/(c)_n>0$ . Then f(z) is starlike of order $\delta$ $(0\leq \delta<1)$ in the disk $|z|< r_2$ , where (4.1) $$r_2 = \inf_{n \in \mathcal{X}} \left[ \frac{(1-\delta)(2+\alpha)_n(a)_n}{(n+1-\delta)(1+\alpha)_n(c)_n} \right]^{1/n}.$$

Corollary 4.1. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(\alpha)$ . Then f(z) is starlike of order $\delta$ (0 $\leq \delta <$ 1) in the disk $|z| < r_3$ , where (4.6) $$r_3 = \inf_{n \in \mathcal{N}} \left[ \frac{(1 - \delta)(2 + \alpha)_n}{(n + 1 - \delta)(1 + \alpha)_n} \right]^{1/n}.$$ Similarly, by applying Theorem 2.4 instead of Theorem 2.3, we have

Theorem 4.2. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(a, c; \alpha)$ with $(a)_n/(c)_n > 0$ . Then f(z) is convex of order $\delta$ $(0 \le \delta < 1)$ in the disk $|z| < r_2$ , where $r_2$ is given by (4.1).

Corollary 4.2. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(\alpha)$ . Then f(z) is convex of order $\delta$ (0 $\leq \delta <$ 1) in the disk $|z| < r_3$ , where $r_3$ is given by (4.6).

Theorem 5.1. Let $(a)_n/(c)_n > 0$ . Then (5.1) $$\mathscr{W}_{0}(a,c;\alpha) \subset \mathscr{V}_{0}\left(a,c;-\frac{2}{3+\alpha}\right).$$ The result is the best possible.

Theorem 5.2. Let (d)J(c)ⁿ > 0. Then, if a^c, $$\mathscr{V}_{\scriptscriptstyle 0}(a,\,c\,;\,\alpha) \subset \mathscr{V}_{\scriptscriptstyle 0}(\alpha)\;,$$ and, if a < c, $$\mathscr{V}_{\scriptscriptstyle 0}(a,c;\alpha)\supset\mathscr{V}_{\scriptscriptstyle 0}(\alpha)\;.$$ Proof Note that (a)J(c)ⁿ is an increasing function of TZ if α ^ c, and is decreasing in 72 if a < c. By using Theorem 2.4 and Corollary 2.6, we complete the proof of Theorem 5.2. Similarly, we have

Theorem 5.3. Let (a)J(c)ⁿ > 0. Then, if a :> c, $$\mathscr{W}_{0}(a,c;\alpha) \subset \mathscr{W}_{0}(\alpha),$$ and, if a < c. $$\mathscr{W}_{0}(a, c; \alpha) \supset \mathscr{W}_{0}(\alpha) .$$

Theorem 6.1. Let the function f(z) be defined by (6.1). Then $f(z) \in \mathscr{V}_k(\alpha, c; \alpha)$ if and only if (6.7) $$\sum_{n=k+1}^{\infty} \frac{(2+\alpha)_n(a)_n}{(1+\alpha)_n(c)_n} a_{n+1} \ge 1 - \sum_{i=1}^k p_i.$$ The result (6.7) is sharp.

Corollary 6.1. Let the function f(z) be defined by (6.1) with a=c. Then $f(z) \in \mathscr{V}_k(\alpha)$ if and only if (6.10) $$\sum_{n=k+1}^{\infty} \frac{(2+\alpha)_n}{(1+\alpha)_n} a_{n+1} \leq 1 - \sum_{i=1}^{k} p_i.$$ The result (6.10) is sharp for the function (6.11) $$f(z) = z - \sum_{i=1}^{k} \frac{(1+\alpha)_i p_i}{(2+\alpha)_i} z^{i+1} - \frac{(1+\alpha)_n \left(1 - \sum_{i=1}^{k} p_i\right)}{(2+\alpha)_n} z^{n+1} \qquad (n \ge k+1).$$

Theorem 6.2. Let the function f(z) be defined by (6.3). Then $f(z) \in \mathcal{W}_k(a, c; \alpha)$ if and only if (6.12) $$\sum_{n=k+1}^{\infty} \frac{(2)_n (2+\alpha)_n (a)_n}{(1)_n (1+\alpha)_n (c)_n} a_{n+1} \leq 1 - \sum_{i=1}^k p_i.$$ The result (6.12) is sharp for the function (6.13) $$f(z) = z - \sum_{i=1}^{k} B_i p_i z^{i+1} - \frac{(1)_n (1 + \alpha)_n (c)_n \left(1 - \sum_{i=1}^{k} p_i\right)}{(2)_n (2 + \alpha)_n (a)_n} z^{n+1} \qquad (n \ge k+1).$$ The proof of Theorem 6.2 is much akin to that of Theorem 6.1, and we omit the details involved.

Corollary 6.2. Let the function f(z) be defined by (6.3) with a = c. Then $f(z) \in \mathcal{W}_k(\alpha)$ if and only if (6.14) $$\sum_{n=k+1}^{\infty} \frac{(2)_n (2+\alpha)_n}{(1)_n (1+\alpha)_n} a_{n+1} \leq 1 - \sum_{i=1}^k p_i.$$ The result (6.14) is sharp for the function (6.15) $$f(z) = z - \sum_{i=1}^{k} \frac{(1)_i (1+\alpha)_i p_i}{(2)_i (2+\alpha)_i} z^{i+1}$$ $$- \frac{(1)_n (1+\alpha)_n \left(1 - \sum_{i=1}^{k} p_i\right)}{(2)_n (2+\alpha)_n} z^{n+1} \qquad (n \ge k+1).$$ With the aid of Theorem 6.1 and Theorem 6.2, we can find the extreme points of the classes $\mathscr{V}_k(a, c; \alpha)$ and $\mathscr{W}_k(a, c; \alpha)$ by using the techniques detailed in Section 3.

Theorem 6.3. The extreme points of the class $\mathscr{V}_k(a,c;\alpha)$ with $(a)_n/(c)_n > 0$ are (6.16) $$f_{k+1}(z) = z - \sum_{i=1}^{k} A_i p_i z^{i+1}$$ and (6.17) $$f_{n+1}(z) = z - \sum_{i=1}^{k} A_i p_i z^{i+1} - \frac{(1+\alpha)_n (c)_n \left(1 - \sum_{i=1}^{k} p_i\right)}{(2+\alpha)_n (a)_n} z^{n+1} \qquad (n \ge k+1),$$ where $A_i$ is given by (6.2).

Theorem 6.4. The extreme points of the class $\mathcal{W}_k(a, c; \alpha)$ with $(a)_n/(c)_n > 0$ are (6.18) $$f_{k+1}(z) = z - \sum_{i=1}^{k} B_i p_i z^{i+1}$$ and (6.19) $$f_{n+1}(z) = z - \sum_{i=1}^{k} B_i p_i z^{i+1} - \frac{(1)_n (1+\alpha)_n (c)_n \left(1 - \sum_{i=1}^{k} p_i\right)}{(2)_n (2+\alpha)_n (a)_n} z^{n+1} \qquad (n \ge k+1),$$ where $B_i$ is given by (6.4).

Theorem 7.1. Let the function f(z) defined by (1.1) by in the class y{a^ c; a). Then, for β ^ a > —1, / {f) is also in the class i^{a, c; a).

Corollary 7.1. Let the function f(z) defined by (1.1) be in the class $\mathscr{V}(a, c; \alpha)$ . Then, for $\beta \geq \alpha > -1$ , (7.14) $$\mathscr{L}(a, c)\mathscr{J}_{\beta}(f) \in \mathscr{S}^ \quad and \quad \mathscr{J}_{\beta}(\mathscr{L}(a, c)f(z)) \in \mathscr{S}^ \ .$$

Theorem 7.2. Let the function f(z) defined by (1.1) be in the class $\mathcal{W}(a, c; \alpha)$ . Then, for $\beta \geq \alpha > -1$ , $\mathcal{J}_{\beta}(f)$ is also in the class $\mathcal{W}(a, c; \alpha)$ .

Corollary 7.2. Let the function f(z) defined by (1.1) be in the class if {a, c; a). Then, for β^a> -1 , (7.18) $$\mathscr{L}(a,c)\mathscr{J}_{\beta}(f)\in\mathscr{K} \quad and \quad \mathscr{J}_{\beta}(\mathscr{L}(a,c)f(z))\in\mathscr{K}.$$

Theorem 8.1. Let the function f(z) defined by (1.26) be in the class $\mathscr{V}_0(2, 2-\lambda; \alpha)$ for $\lambda < 1$ . Then $$|D_z^{\lambda}f(z)| \geq \frac{|z|^{1-\lambda}}{\Gamma(2-\lambda)} \left\{ 1 - \left(\frac{1+\alpha}{2+\alpha}\right)|z| \right\}$$ and $$|D_z^{\lambda}f(z)| \leq \frac{|z|^{1-\lambda}}{\Gamma(2-\lambda)} \left\{ 1 + \left(\frac{1+\alpha}{2+\alpha}\right)|z| \right\}$$ for $z \in \mathcal{U}$ . The results (8.9) and (8.10) are sharp.

Theorem 8.2. Let the function f(z) defined by (1.26) be in the class $\mathcal{W}_0(2, 2-\lambda; \alpha)$ for $\lambda < 1$ . Then $$|D_z^{\lambda}f(z)| \geq \frac{|z|^{1-\lambda}}{\Gamma(2-\lambda)} \left(1 - \frac{1+\alpha}{2(2+\alpha)}|z|\right)$$ and $$|D_z^{\lambda}f(z)| \leq \frac{|z|^{1-\lambda}}{\Gamma(2-\lambda)} \left(1 + \frac{1+\alpha}{2(2+\alpha)}|z|\right)$$ for $z \in \mathcal{U}$ . The results (8.13) and (8.14) are sharp for the function (8.15) $$f(z) = z - \frac{(1+\alpha)(2-\lambda)}{4(2+\alpha)}z^2.$$

Theorem 8.3. Let $\lambda < 1$ . Then (8.16) $$\Omega^{1+\lambda} \mathscr{W}(2, 2-\lambda; \alpha) \subset \mathscr{V}(\alpha).$$

Corollary 8.1. Let $\lambda < 1$ . Then (8.19) $$\Omega^{1+\lambda} \mathscr{W}_{0}(2, 2-\lambda; \alpha) \subset \mathscr{V}_{0}(\alpha).$$

Corollary 8.2. Let $\lambda < 1$ . Then (8.20) $$\Omega^{1+\lambda} \mathscr{W}_{k}(2, 2-\lambda; \alpha) \subset \mathscr{V}_{k}(\alpha).$$ Next we prove

Theorem 8.4. Let $\lambda < 1$ . Then (8.21) $$\Omega^{\lambda} \mathscr{W}(1, 2 - \lambda; \alpha) \subset \mathscr{V}(\alpha).$$

Corollary 8.3. Let $\lambda < 1$ . Then (8.23) $$\Omega^{\lambda} \mathscr{W}_{0}(1, 2 - \lambda; \alpha) \subset \mathscr{V}_{0}(\alpha).$$

Corollary 8.4. Let $\lambda < 1$ . Then (8.24) $$\Omega^{\lambda} \mathscr{W}_{k}(1, 2-\lambda; \alpha) \subset \mathscr{V}_{k}(\alpha).$$ Finally, we prove the following theorem involving generalized hypergeometric functions.

Theorem 8.5. Let $\lambda < 1$ . Then $$(8.25) z_{q+2}F_{q+1}(2, \cdots, 2, 2; 1, \cdots, 1, 2-\lambda; z) \in \Omega^{q+1+\lambda}(\mathscr{S}^*(\frac{1}{2}) \cap \mathscr{K}).$$ Proof. Note that (8.26) $$z_{q+2}F_{q+1}(2, \dots, 2, 2; 1, \dots, 1, 2 - \lambda; z)$$ $$= \mathcal{L}^{q+1}(2, 1)\mathcal{L}(2, 2 - \lambda)\phi(1, 1; z)$$ $$= \Omega^{q+1+\lambda}\phi(1, 1; z) ,$$ where $\phi(1, 1; z)$ is defined by (1.10). It is easy to show that $$\phi(1, 1; z) \in \mathcal{S}^*(\frac{1}{2}) \cap \mathcal{K},$$ which obviously completes the proof of Theorem 8.5.

Definition 8.1. The fractional integral of order λ is defined, for a function f(z), by (8.1) $$D_z^{-\lambda} f(z) = \frac{1}{\Gamma(\lambda)} \int_0^z \frac{f(\zeta)}{(z-\zeta)^{1-\lambda}} d\zeta,$$ where λ > 0, f(z) is an analytic function in a simply-connected region of the 2-plane containing the origin, and the multiplicity of (z — ζ)^"¹ is removed by requiring log (z — ζ) to be real when z — ζ > 0.

Definition 8.2. The fractional derivative of order λ is defined, for a function f(z), by (8.2) $$D_z^{\lambda} f(z) = \frac{1}{\Gamma(1-\lambda)} \frac{d}{dz} \int_0^z \frac{f(\zeta)}{(z-\zeta)^{\lambda}} d\zeta,$$ where 0 ^ λ < 1, f(z) is an analytic function in a simply-connected region of the £-plane containing the origin, and the multiplicity of (z — ζ)~ is removed as in Definition 8.1 above.

Definition 8.3. Under the hypotheses of Definition 8.2, the fractional derivative of order n + λ is defined by $$(8.3) D_z^{n+\lambda}f(z) = \frac{d^n}{dz^n} D_z^{\lambda}f(z) ,$$ where $0 \leq \lambda < 1$ , and $n \in \mathcal{N} \cup \{0\}$ . Following the lines of Owa and Srivastava [9], we introduce the linear operator $\Omega^{\lambda}$ defined by (8.4) $$\Omega^{\lambda} f = \Gamma(2-\lambda) z^{\lambda} D_{z}^{\lambda} f(z) \qquad (\lambda < 1)$$ for functions f(z) belonging to the class $\mathcal{A}$ . Note that (8.5) $$\Omega^{\lambda} f = \mathcal{L}(2, 2 - \lambda) f \qquad (\lambda < 1)$$ for functions f(z) defined by (1.1). Further, we define (8.6) $$\Omega f \equiv \Omega^1 f = z f'(z) = \mathcal{L}(2, 1) f$$ and (8.7) $$\Omega^n f = \Omega(\Omega^{n-1} f) = \mathcal{L}^n(2, 1) f \qquad (n \in \mathcal{N}),$$ with, of course, $\Omega^0 f = f(z)$ . Then it is easily observed that (8.8) $$\Omega^{n+\lambda}f = \Omega^n(\Omega^{\lambda}f) = \mathcal{L}^n(2,1)\mathcal{L}(2,2-\lambda)f = \Omega^{\lambda}(\Omega^nf)$$ for $\lambda < 1$ and $n \in \mathcal{N}$ . We begin by proving