Li et al. Journal of Inequalities and Applications 2013, 2013:86

Theorem 2.1. Let the function $f \in A_p$ and suppose that $g \in L^{j,l}_{p,q}[A,B;\alpha,\gamma]$ . If $D^j f^{(q)}(z)$ is majorized by $D^l g^{(q)}(z)$ in $\Delta$ , and <span id="page-2-1"></span> $$(p-q)^{j-l} \geq \left[\frac{(A-B)|\gamma|}{1-\alpha} + (p-q)^{j-l}|B|\right]\delta,$$ <span id="page-2-0"></span>then $$|D^{j+1}f^{(q)}(z)| \le |D^{l+1}g^{(q)}(z)| \quad (|z| \le r_0),$$ (2.1) where $r_0 = r_0(p, q, \alpha, \gamma, j, l, A, B)$ is the smallest positive root of the equation $$\left[ \frac{(A-B)|\gamma|}{1-\alpha} + (p-q)^{j-l}|B| \right] r^{3} - \left[ (p-q)^{j-l} + 2|B| \right] r^{2} - \left[ \frac{(A-B)|\gamma|}{1-\alpha} + (p-q)^{j-l}|B| + 2 \right] r + (p-q)^{j-l} = 0 \left( -1 \le B < A \le 1; p, j \in N; q, l \in N_{0}; 0 \le \alpha < 1; \gamma \in C^{*}, 0 \le \delta \le r_{0} \right).$$ (2.2) Proof Suppose that $g \in L^{j,l}_{p,q}[A,B;\alpha,\gamma]$ . Then, making use of the fact that $$\varpi - \alpha |\varpi - 1| \prec \frac{1 + Az}{1 + Bz} \quad \Leftrightarrow \quad \varpi \left( 1 - \alpha e^{-i\phi} \right) + \alpha e^{-i\phi} \prec \frac{1 + Az}{1 + Bz} \quad (\phi \in R),$$ and letting $$\varpi = 1 + \frac{1}{\gamma} \left( \frac{D^j g^{(q)}(z)}{D^l g^{(q)}(z)} - (p - q)^{j-l} \right)$$ in (1.7), we obtain $$\left[1+\frac{1}{\gamma}\left(\frac{D^{j}g^{(q)}(z)}{D^{l}g^{(q)}(z)}-(p-q)^{j-l}\right)\right]\left(1-\alpha e^{-i\phi}\right)+\alpha e^{-i\phi}\prec\frac{1+Az}{1+Bz}$$ or, equivalently, <span id="page-3-1"></span><span id="page-3-0"></span> $$1 + \frac{1}{\gamma} \left( \frac{D^{j} g^{(q)}(z)}{D^{l} g^{(q)}(z)} - (p - q)^{j - l} \right) < \frac{1 + (\frac{A - \alpha B e^{-i\phi}}{1 - \alpha e^{-i\phi}}) z}{1 + B z}$$ (2.3) which holds true for all $z \in \Delta$ . We find from (2.3) that $$1 + \frac{1}{\gamma} \left( \frac{D^{j} g^{(q)}(z)}{D^{l} g^{(q)}(z)} - (p - q)^{j - l} \right) = \frac{1 + (\frac{A - \alpha B e^{-i\phi}}{1 - \alpha e^{-i\phi}})\omega(z)}{1 + B\omega(z)}, \tag{2.4}$$ where $\omega(z) = c_1 z + c_2 z^2 + \cdots$ , $\omega \in P$ , P denotes the well-known class of the bounded analytic functions in $\Delta$ and satisfies the conditions <span id="page-3-2"></span> $$\omega(0) = 0$$ and $|\omega(z)| \le |z|$ $(z \in \Delta)$ . <span id="page-3-3"></span>From (2.4), we get $$\frac{D^{j}g^{(q)}(z)}{D^{l}g^{(q)}(z)} = \frac{(p-q)^{j-l} + \left[\frac{(A-B)\gamma}{1-\alpha e^{-i\phi}} + (p-q)^{j-l}B\right]\omega(z)}{1+B\omega(z)}.$$ (2.5) By virtue of (2.5), we obtain $$\left| D^{l} g^{(q)}(z) \right| \leq \frac{1 + |B||z|}{(p - q)^{j-l} - \left| \frac{(A - B)\gamma}{1 - \alpha e^{-i\phi}} + (p - q)^{j-l} B||z| \right|} \left| D^{j} g^{(q)}(z) \right| \leq \frac{1 + |B||z|}{(p - q)^{j-l} - \left[ \frac{(A - B)|\gamma|}{1 - \alpha} + (p - q)^{j-l} |B| \right]|z|} \left| D^{j} g^{(q)}(z) \right|.$$ (2.6) Next, since $D^i f^{(q)}(z)$ is majorized by $D^l g^{(q)}(z)$ in $\Delta$ , thus from (1.3), we have <span id="page-3-5"></span> $$D^{j}f^{(q)}(z) = \varphi(z)D^{l}g^{(q)}(z).$$ Differentiating the above equality with respect to z and multiplying by z, we get <span id="page-3-4"></span> $$D^{j+1}f^{(q)}(z) = z\varphi'(z)D^{l}g^{(q)}(z) + \varphi(z)D^{l+1}g^{(q)}(z).$$ (2.7) Thus, by noting that $\varphi(z) \in P$ satisfies the inequality (see, e.g., Nehari [21]) $$\left|\varphi'(z)\right| \le \frac{1 - |\varphi(z)|^2}{1 - |z|^2} \quad (z \in \Delta) \tag{2.8}$$ and making use of (2.6) and (2.8) in (2.7), we obtain $$\left| D^{j+1} f^{(q)}(z) \right| \le \left( \left| \varphi(z) \right| + \frac{1 - |\varphi(z)|^2}{1 - |z|^2} \cdot \frac{(1 + |B||z|)|z|}{[(p - q)^{j-l} - (\frac{(A - B)|\gamma|}{1 - \alpha} + (p - q)^{j-l}|B|)|z|]} \right) \times \left| D^{l+1} g^{(q)}(z) \right|, \tag{2.9}$$ which, upon setting $$|z| = r$$ and $|\varphi(z)| = \rho$ $(0 \le \rho \le 1)$ , leads us to the inequality $$\begin{split} & \left| D^{j+1} f^{(q)}(z) \right| \\ & \leq \left( \frac{\psi(\rho)}{(1-r^2)[(p-q)^{j-l} - (\frac{(A-B)|\gamma|}{1-\alpha} + (p-q)^{j-l}|B|)r]} \right) \left| D^{l+1} g^{(q)}(z) \right|, \end{split}$$ where $$\psi(\rho) = -r(1+|B|r)\rho^{2} + (1-r^{2})\left[(p-q)^{j-l} - \left(\frac{(A-B)|\gamma|}{1-\alpha} + (p-q)^{j-l}|B|\right)r\right]\rho + r(1+|B|r)$$ (2.10) <span id="page-4-0"></span>takes its maximum value at $\rho = 1$ with $r_0 = r_0(p, q, \alpha, \gamma, j, l, A, B)$ , where $$r_0 = r_0(p, q, \alpha, \gamma, j, l, A, B)$$ is the smallest positive root of equation (2.2). Furthermore, if $0 \le \delta \le r_0(p, q, \alpha, \gamma, j, l, A, B)$ , then the function $\psi(\rho)$ defined by $$\psi(\rho) = -\delta (1 + |B|\delta) \rho^{2} + (1 - \delta^{2}) \left[ (p - q)^{j-l} - \left( \frac{(A - B)|\gamma|}{1 - \alpha} + (p - q)^{j-l} |B| \right) \delta \right] \rho + \delta (1 + |B|\delta)$$ (2.11) is an increasing function on the interval $0 \le \rho \le 1$ so that $$\psi(\rho) \le \psi(1) = \left(1 - \delta^{2}\right) \left[ (p - q)^{j-l} - \left(\frac{(A - B)|\gamma|}{1 - \alpha} + (p - q)^{j-l}|B|\right) \delta \right]$$ $$(0 \le \rho \le 1; 0 \le \delta \le r_{0}(p, q, \alpha, \gamma, j, l, A, B)).$$ (2.12) <span id="page-4-1"></span>Hence, upon setting $\rho = 1$ in (2.11), we conclude that (2.1) of Theorem 2.1 holds true for $|z| \le r_0(p, q, \alpha, \gamma, j, l, A, B)$ , which completes the proof of Theorem 2.1. Setting $\alpha = 0$ in Theorem 2.1, we get the following result.

Corollary 2.1. Let the function $f \in A_p$ and suppose that $g \in S_{p,q}^{j,l}[A,B;\gamma]$ . If $D^j f^{(q)}(z)$ is majorized by $D^l g^{(q)}(z)$ in $\Delta$ , and $$(p-q)^{j-l} \ge \left[ (A-B)|\gamma| + (p-q)^{j-l}|B| \right] \delta,$$ then $$|D^{j+1}f^{(q)}(z)| \le |D^{l+1}g^{(q)}(z)| \quad (|z| \le r_0),$$ (2.13) where $r_0 = r_0(p, q, \gamma, j, l, A, B)$ is the smallest positive root of the equation $$[(A-B)|\gamma| + (p-q)^{j-l}|B|]r^{3} - [(p-q)^{j-l} + 2|B|]r^{2} - [(A-B)|\gamma| + (p-q)^{j-l}|B| + 2]r$$ $$+ (p-q)^{j-l} = 0$$ $$(-1 \le B < A \le 1; p, j \in N; q, l \in N_{0}; \gamma \in C^{*}, 0 \le \delta \le r_{0}).$$ (2.14) <span id="page-5-0"></span>Remark 2.1 Corollary 2.1 improves the result of Goswami and Aouf [4, Theorem 1]. Putting p = 1, q = 0, j = m, l = n, m > n and $\gamma = 1$ in Theorem 2.1, we obtain the following result.

Corollary 2.2. Let the function $f \in A$ and suppose that $g \in U_{m,n}(\alpha, A, B)$ . If $D^m f(z)$ is majorized by $D^n g(z)$ in $\Delta$ , then $$|D^{m+1}f(z)| \le |D^{n+1}g(z)| \quad (|z| \le r_0),$$ (2.15) where $r_0 = r_0(\alpha, A, B)$ is the smallest positive root of the equation <span id="page-5-1"></span> $$\left[\frac{A-B}{1-\alpha} + |B|\right]r^3 - \left(1+2|B|\right)r^2 - \left[\frac{A-B}{1-\alpha} + |B| + 2\right]r + 1 = 0$$ $$(-1 \le B < A \le 1; 0 \le \alpha < 1). \tag{2.16}$$ For $A = 1 - 2\beta$ , B = -1, putting m = 1, n = 0 and m = 2, n = 1 in Corollary 2.2, respectively, we obtain the following Corollaries 2.3 and 2.4.

Corollary 2.3. Let the function $f \in A$ and suppose that $g \in US(\alpha, \beta)$ . If Df(z) is majorized by g(z) in $\Delta$ , then $$|f'(z)+zf''(z)| \leq |g'(z)| \quad (|z| \leq r_0),$$ <span id="page-5-2"></span>where $r_0 = r_0(\alpha, \beta)$ is the smallest positive root of the equation $$\left[\frac{2(1-\beta)}{1-\alpha} + 1\right]r^3 - 3r^2 - \left[\frac{2(1-\beta)}{1-\alpha} + 3\right]r + 1 = 0 \quad (0 \le \alpha < 1; 0 \le \beta < 1).$$

Corollary 2.4. Let the function $f \in A$ and suppose that $g \in UK(\alpha, \beta)$ . If $D^2f(z)$ is majorized by Dg(z) in $\Delta$ , then $$|D^3 f(z)| \le |D^2 g(z)| \quad (|z| \le r_0),$$ where $r_0 = r_0(\alpha, \beta)$ is the smallest positive root of the equation $$\left[\frac{2(1-\beta)}{1-\alpha} + 1\right]r^3 - 3r^2 - \left[\frac{2(1-\beta)}{1-\alpha} + 3\right]r + 1 = 0 \quad (0 \le \alpha < 1; 0 \le \beta < 1).$$ Also, putting A = 1, B = -1, q = 0, j = n + 1 and l = n in Theorem 2.1, we obtain the following result.

Corollary 2.5. Let the function $f \in A_p$ and suppose that $g \in S_n(p,\alpha,\gamma)$ . If $D^{n+1}f(z)$ is majorized by $D^ng(z)$ in $\Delta$ , then $$|D^{n+2}f(z)| \le |D^{n+1}g(z)| \quad (|z| \le r_0),$$ (2.17) where $r_0 = r_0(p, \alpha, \gamma)$ is the smallest positive root of the equation <span id="page-6-1"></span> $$\left[\frac{2|\gamma|}{1-\alpha} + p\right]r^{3} - (p+2)r^{2} - \left[\frac{2|\gamma|}{1-\alpha} + p+2\right]r + p = 0$$ $$(p \in N; \gamma \in C^{*}; 0 \le \alpha < 1). \tag{2.18}$$

Definition 1.1. A function $f(z) \in A_p$ is said to be in the class $L_{p,q}^{j,l}[A,B;\alpha,\gamma]$ of p-valent functions of complex order $\gamma \neq 0$ in $\Delta$ if and only if $$\left[1 + \frac{1}{\gamma} \left( \frac{D^{j} f^{(q)}(z)}{D^{l} f^{(q)}(z)} - (p - q)^{j - l} \right) - \alpha \left| \frac{1}{\gamma} \left( \frac{D^{j} f^{(q)}(z)}{D^{l} f^{(q)}(z)} - (p - q)^{j - l} \right) \right| \right] < \frac{1 + Az}{1 + Bz} \left(z \in \Delta; -1 \le B < A \le 1; j > l; p, j \in N; l, q \in N_0; 0 \le \alpha; \gamma \in C^* = C \setminus \{0\}\right).$$ (1.7) Clearly, we have the following relationships: - (1) $L_{p,q}^{j,l}[A,B;0,\gamma] = S_{p,q}^{j,l}[A,B;\gamma];$ - (2) $L_{1,0}^{m,n}[A,B;\alpha,1]=U_{m,n}(\alpha,A,B);$ - (3) $L_{1,0}^{1,0}[1-2\beta,-1;\alpha,1]=US(\alpha,\beta)$ ( $0\leq \beta<1$ ) ( $\alpha$ -uniformly starlike functions of order $\beta$ ); - (4) $L_{2,1}^{1,0}[1-2\beta,-1;\alpha,1]=UK(\alpha,\beta)$ ( $0\leq \beta<1$ ) ( $\alpha$ -uniformly convex functions of order $\beta$ ); - (5) $L_{n,0}^{n+1,n}[1,-1;\alpha,\gamma] = S_n(p,\alpha,\gamma) \ (n \in N_0);$ - (6) $L_{1,0}^{1,0}[1,-1;\alpha,\gamma] = S(\alpha,\gamma) \ (0 \le \alpha < 1, \gamma \in C^*);$ - (7) $L_{1,0}^{2,1}[1,-1;\alpha,\gamma] = K(\alpha,\gamma) \ (0 \le \alpha < 1, \gamma \in C^*);$ - (8) $L_{1,0}^{1,0}[1,-1;\alpha,1-\beta] = S^*(\alpha,\beta) \ (0 \le \alpha < 1, \ 0 \le \beta < 1).$ The classes $S_{p,q}^{j,l}[A,B;\gamma]$ and $U_{m,n}(\alpha,A,B)$ were introduced by Goswami and Aouf [4] and Li and Tang [5], respectively. The classes $US(\alpha,\beta)$ and $UK(\alpha,\beta)$ were studied recently in [6] (see also [7–12]). The class $S_n(p,0,\gamma) = S_n(p,\gamma)$ was introduced by Akbulut et al. [13]. Also, the classes $S(0,\gamma) = S(\gamma)$ and $K(0,\gamma) = K(\gamma)$ are said to be classes of starlike and convex of complex order $\gamma \neq 0$ in $\Delta$ which were considered by Nasr and Aouf [14] and Wiatrowski [15] (see also [16, 17]), and $S^(0,\beta) = S^(\beta)$ denotes the class of starlike functions of order $\beta$ in $\Delta$ . A majorization problem for the class $S(\gamma)$ has recently been investigated by Altintas et al. [18]. Also, majorization problems for the classes $S^*(\beta)$ and $S_{p,q}^{j,l}[A,B;\gamma]$ have been investigated by MacGregor [1] and Goswami and Aouf [4], respectively. Very recently, Goyal and Goswami [19] (see also [20]) generalized these results for the fractional derivative operator. In the present paper, we investigate a majorization problem for the class $L_{p,q}^{j,l}[A,B;\alpha,\gamma]$ .