Ul-Haq and Noor Journal of Inequalities and Applications 2012, 2012:309

Lemma 1.1. A function $f \in V_k(\eta, \rho)$ if and only if - (i) $f'(z) = [f'_1(z)]^{(1-\rho)\eta}, f_1(z) \in V_k$ - (ii) $f'(z) = [f_2'(z)]^{\eta}, f_1(z) \in V_k(\rho),$ - <span id="page-1-2"></span>(iii) there exist two normalized starlike functions $s_1(z)$ and $s_2(z)$ such that $$f'(z) = \left[ \frac{(s_1(z)/z)^{(\frac{k}{4} + \frac{1}{2})}}{(s_2(z)/z)^{(\frac{k}{4} - \frac{1}{2})}} \right]^{(1-\rho)\eta}.$$ (1.2) <span id="page-1-3"></span>The above lemma is a special case of the result discussed in [6].

Lemma 1.2. [7] Let $h \in P$ with $z = re^{i\theta}$ . Then $$\frac{1}{2\pi} \int_{0}^{2\pi} \left| h(z) \right|^{2} d\theta \leq \frac{1 + 3r^{2}}{1 - r^{2}}.$$

Lemma 1.3. [8] Let f be univalent and $0 \le r < 1$ . Then there exists a number $z_1$ with $|z_1| = r$ , such that for all z, |z| = r, we have $$|z-z_1| |f(z)| \leq \frac{2r^2}{1-r^2}.$$

Theorem 2.1. The function $f \in \widetilde{N}_k(\eta, \rho, \beta)$ if and only if <span id="page-2-2"></span> $$f'(z) = \frac{(f_1(z))^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\eta}}{(f_2(z))^{(\frac{k}{4} - \frac{1}{2})(1-\rho)\eta}},$$ where $f_1$ and $f_2$ are strongly close-to-convex functions of order $\beta$ .

Theorem 2.2. Let $f \in \widetilde{N}_k(\eta, \rho, \beta)$ in E. Then $f \in C_n$ for $|z| < r_0$ , where $$r_0 = \frac{2|\eta|}{[(1-\rho)|\eta|k+2\beta] + \sqrt{[(1-\rho)|\eta|k+2\beta]^2 - 4(1-2\rho)|\eta|^2}}.$$ (2.1) This result is sharp. Proof We can write <span id="page-2-0"></span> $$f'(z) = g'(z)h^{\beta}(z), \quad g(z) \in V_k(\eta, \rho), h(z) \in P$$ Using Lemma 1.1, we have $$f'(z) = \left[ \frac{(s_1(z)/z)^{(\frac{k}{4} + \frac{1}{2})}}{(s_2(z)/z)^{(\frac{k}{4} - \frac{1}{2})}} \right]^{(1-\rho)\eta} h^{\beta}(z), \tag{2.2}$$ where $s_1$ and $s_2$ are starlike functions. Logarithmic differentiation of (2.2) gives us $$\frac{zf''(z)}{f'(z)} = (1-\rho)\eta \left[ -1 + \left(\frac{k}{4} + \frac{1}{2}\right) \frac{zs_1'(z)}{s_1(z)} - \left(\frac{k}{4} - \frac{1}{2}\right) \frac{zs_2'(z)}{s_2(z)} \right] + \beta \frac{zh'(z)}{h(z)},$$ <span id="page-2-1"></span>which implies that $$1 + \frac{1}{\eta} \frac{zf''(z)}{f'(z)} = \rho + (1 - \rho) \left\lceil \left(\frac{k}{4} + \frac{1}{2}\right) \frac{zs_1'(z)}{s_1(z)} - \left(\frac{k}{4} - \frac{1}{2}\right) \frac{zs_2'(z)}{s_2(z)} \right\rceil + \frac{\beta}{\eta} \frac{zh'(z)}{h(z)}.$$ Now using distortion results for the class P, we have $$\operatorname{Re}\left(1 + \frac{1}{\eta} \frac{zf''(z)}{f'(z)}\right) \ge \rho + (1 - \rho) \left[ \left(\frac{k}{4} + \frac{1}{2}\right) \frac{1 - r}{1 + r} - \left(\frac{k}{4} - \frac{1}{2}\right) \frac{1 + r}{1 - r} \right] - \frac{\beta}{|\eta|} \frac{2r}{1 - r^2}$$ $$= \frac{\rho |\eta|(1 - r^2) + (1 - \rho)|\eta|[1 - kr + r^2] - 2\beta r}{|\eta|(1 - r^2)}.$$ (2.3) The right-hand side of (2.3) is positive for $|z| < r_0$ , where $r_0$ is given by (2.1). The sharpness can be viewed from the function $f_0 \in \widetilde{N}_k(\eta, \rho, \beta)$ , given by $$f_0'(z) = \frac{(1+z)^{(\frac{k}{2}-1)(1-\rho)\eta+\beta}}{(1-z)^{(\frac{k}{2}+1)(1-\rho)\eta+\beta}}, \quad z \in E.$$ (2.4) We note the following. - (i) For $\eta = 1$ , we have the radius of convexity for the class $\widetilde{T}_k(\rho, \beta)$ studied by Noor [4]. - (ii) For $\eta = 1$ , $\rho = 0$ , $\beta = 1$ , we have the radius of convexity for the class $T_k$ , proved by Noor [3]. - <span id="page-3-0"></span>(iii) For $\eta = 1$ , $\rho = 0$ , $\beta = 1$ , k = 2, we have the radius of convexity for close-to-convex functions which is well known. We now discuss the arc length problem and the growth rate of coefficients for the class $\widetilde{N}_k(\eta, \rho, \beta)$ .

Theorem 2.3. Let $f \in \widetilde{N}_k(\eta, \rho, \beta)$ , for $\operatorname{Re} \eta > 0$ , $\beta \geq 0$ , $0 \leq \rho < 1$ and $\frac{(k+2)(1-\rho)\operatorname{Re} \eta}{2-\beta} > 1$ . Then $$L_r(f) \leq c(k,\eta,\rho,\beta) \left(\frac{1}{1-r}\right)^{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta+\beta-1},$$ where $c(k, \eta, \rho, \beta)$ is a constant depending only on k, $\eta$ , $\rho$ , $\beta$ . The exponent $[(\frac{k}{2} + 1)(1 - \rho) \operatorname{Re} \eta + \beta - 1]$ is sharp. Proof We have $$L_r(f) = \int_0^{2\pi} |zf'(z)| d\theta, \quad z = re^{i\theta}.$$ Using Definition 1.1, Lemma 1.1(iii) and the distortion theorem for starlike functions, we have $$\begin{split} L_r(f) &= \int_0^{2\pi} \left| z g'(z) h^{\beta}(z) \right| d\theta, \quad g(z) \in V_k(\eta, \rho), h(z) \in P \\ &= \int_0^{2\pi} \left| z \frac{(s_1(z)/z)^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\eta}}{(s_2(z)/z)^{(\frac{k}{4} - \frac{1}{2})(1-\rho)\eta}} \right| \left| h^{\beta}(z) \right| d\theta \\ &= \int_0^{2\pi} \left| z^{1-\eta(1-\rho)} \frac{(s_1(z))^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\eta}}{(s_2(z))^{(\frac{k}{4} - \frac{1}{2})(1-\rho)\eta}} \right| \left| h^{\beta}(z) \right| d\theta \\ &\leq \frac{2^{(\frac{k}{2} - 1)(1-\rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\operatorname{Re}\eta} - 1} \int_0^{2\pi} \left| s_1(z) \right|^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\operatorname{Re}\eta} \left| h(z) \right|^{\beta} d\theta \\ &\leq \frac{2^{(\frac{k}{2} - 1)(1-\rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4} + \frac{1}{2})(1-\rho)\operatorname{Re}\eta - 1}} \int_0^{2\pi} \left( \left| s_1(z) \right|^{\frac{(\frac{k}{2} + 1)(1-\rho)\operatorname{Re}\eta}{2-\beta}} \right)^{\frac{2-\beta}{2}} \left( \left| h(z) \right|^2 \right)^{\frac{\beta}{2}} d\theta \,. \end{split}$$ Using Holder's inequality with $p = \frac{2}{2-\beta}$ , $q = \frac{2}{\beta}$ such that $\frac{1}{p} + \frac{1}{q} = 1$ , we obtain $$\begin{split} L_r(f) &\leq \frac{2^{(\frac{k}{2}-1)(1-\rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-1}} \bigg(\frac{1}{2\pi} \int_0^{2\pi} \left|s_1(z)\right|^{\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta}{2-\beta}} d\theta\bigg)^{\frac{2-\beta}{2}} \\ &\times \bigg(\frac{1}{2\pi} \int_0^{2\pi} \left|h(z)\right|^2 d\theta\bigg)^{\frac{\beta}{2}}. \end{split}$$ <span id="page-4-0"></span> Since $\frac{(k+2)(1-\rho)\operatorname{Re}\eta}{2-\beta} > 1$ , therefore using subordination for starlike functions and Lemma 1.2, we have $$\begin{split} L_r(f) &\leq \frac{2^{(\frac{k}{2}-1)(1-\rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-1}} \left(\frac{1}{2\pi} \int_0^{2\pi} \left|s_1(z)\right|^{\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta}{2-\beta}} d\theta\right)^{\frac{2-\beta}{2}} \left(\frac{1+3r^2}{1-r^2}\right)^{\frac{\beta}{2}} \\ &\leq \frac{2^{(\frac{k}{2}-1)(1-\rho)\operatorname{Re}\eta+\frac{\beta}{2}}}{r^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-1}} \left(\frac{1}{1-r}\right)^{\frac{\beta}{2}} \left(\frac{1}{2\pi} \int_0^{2\pi} \frac{r^{\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta}{2-\beta}}}{|1+re^{i\theta}|^{\frac{(k+2)(1-\rho)\operatorname{Re}\eta}{2-\beta}}} d\theta\right)^{\frac{2-\beta}{2}} \\ &\leq \frac{2^{(\frac{k}{2}-1)(1-\rho)\operatorname{Re}\eta+\frac{\beta}{2}}}{r^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-1-\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta}{2-\beta}}} \left(\frac{1}{1-r}\right)^{\frac{\beta}{2}} \left(\frac{1}{2\pi} \int_0^{2\pi} \frac{1}{|1+re^{i\theta}|^{\frac{(k+2)(1-\rho)\operatorname{Re}\eta}{2-\beta}}} d\theta\right)^{\frac{2-\beta}{2}} \\ &\leq c(k,\eta,\rho,\beta) \left(\frac{1}{1-r}\right)^{\frac{\beta}{2}} \left(\frac{1}{1-r}\right)^{\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta+\beta-1}}{(\frac{k^2}{1-r})^{\frac{\beta}{2}}} \left(\frac{1}{1-r}\right)^{\frac{\beta}{2}} \left(\frac{1}{1-r}\right)^{\frac{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta+\beta-1}}{(\frac{k^2}{1-r})^{\frac{\beta}{2}}} \right). \end{split}$$ The function $F_0 \in \widetilde{N}_k(\eta, \rho, \beta)$ defined by $$F_0'(z) = G_0'(z)h_0^{\beta}(z), \tag{2.5}$$ where $$G_0'(z) = \frac{(1+z)^{(\frac{k}{2}-1)(1-\rho)\eta}}{(1-z)^{(\frac{k}{2}+1)(1-\rho)\eta}} \quad \text{and} \quad h_0(z) = \frac{1+z}{1-z},$$ shows that the exponent is sharp. By assigning different values to the parameters involved in the above theorem, we have the following interesting results.

Corollary 2.1. Let $f \in \widetilde{T}_k(\rho, \beta)$ . Then $$L_r(f) \le c(k, \rho, \beta) \left(\frac{1}{1-r}\right)^{(\frac{k}{2}+1)(1-\rho)+\beta-1}.$$

Corollary 2.2. Let $f \in T_k$ . Then $$L_r(f) \le c(k, \rho, \beta) \left(\frac{1}{1-r}\right)^{\frac{k}{2}+1}.$$ Coefficient growth problems The problem of growth rate and asymptotic behavior of coefficients is well known. In the upcoming results, we investigate these problems for a different set of classes by varying different parameters.

Theorem 2.4. Let $f \in \widetilde{N}_k(\eta, \rho, \beta)$ and be of the form (1.1). Then, for n > 3, $k \ge 2$ , $\text{Re } \eta > 0$ , $0 < \rho < 1$ , $\beta > 0$ , we have $$|a_n| \leq c(k, \eta, \rho, \beta) n^{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta + \beta - 2},$$ where $c(k, \eta, \rho, \beta)$ is a constant depending only on k, $\eta$ , $\rho$ , $\beta$ . The exponent $[(\frac{k}{2} + 1)(1 - \rho) \operatorname{Re} \eta + \beta - 2]$ is sharp. Proof With $z = re^{i\theta}$ , Cauchy's theorem gives us $$na_n = \frac{1}{2\pi r^n} \int_0^{2\pi} |zf'(z)| d\theta = \frac{1}{2\pi r^n} L_r(f), \quad z = re^{i\theta}.$$ Using Theorem 2.3 and putting $r = 1 - \frac{1}{n}$ , we obtain the required result. The sharpness follows from the function $F_0$ defined by the relation (2.5).

Corollary 2.3. Let $f \in \widetilde{T}_k(\rho, \beta)$ and be of the form (1.1). Then, for n > 3, $k \ge 2$ , we have <span id="page-5-0"></span> $$|a_n| = O(1)n^{(\frac{k}{2}+1)(1-\rho)+\beta-2}$$ . For $\rho = 0$ , $\beta = 1$ in the above corollary, we have the growth rate of coefficients problem for functions in this class $T_k$ and for k = 2, $\rho = 0$ , $\beta = 1$ gives us the growth rate of coefficient estimates for close-to-convex functions, which is well known.

Lemma 3.1. Let $f \in A$ and let the Hankel determinant of f be defined by (3.1). Then, writing $\Delta_i = \Delta_i(n, z_1, f)$ , we have <span id="page-6-1"></span> $$H_{q}(n) = \begin{vmatrix} \Delta_{2q-2}(n) & \Delta_{2q-3}(n+1) & \cdots & \Delta_{q-1}(n+q-1) \\ \Delta_{2q-3}(n+1) & \Delta_{2q-4}(n+2) & \cdots & \Delta_{q-2}(n+q) \\ \vdots & \vdots & \vdots & \vdots \\ \Delta_{q-1}(n+q-1) & \Delta_{q-2}(n+q) & \cdots & \Delta_{0}(n+2q-2) \end{vmatrix}.$$ (3.2)

Lemma 3.2. With $z_1 = \frac{n}{n+1}y$ and $v \ge 0$ any integer, $$\Delta_j (n+\nu, z_1, zf') = \sum_{l=0}^j \binom{j}{l} \frac{y^l (\nu - (l-1)n)}{(n+1)^l} \Delta_{j-l} (n+\nu + l, y, f).$$ We also need the following remark given in [10].

Theorem 3.1. Let $f \in \widetilde{N}_k(\eta, \rho, \beta)$ and let the Hankel determinant of f, for $q \ge 2$ , $n \ge 1$ , be defined by (3.1). Then, for $q \ge 2$ and $k > 4\frac{(q-1)}{(1-\rho)\operatorname{Re}\eta} - 2$ , we have $$H_q(n) = O(1)n^{[(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta]q-q^2-(n+1)q},$$ where O(1) depends only on k, $\eta$ , $\rho$ , $\beta$ and q. Proof Since $f \in \widetilde{N}_k(\eta, \rho, \beta)$ , there exists $g \in V_k(\eta, \rho)$ such that $$f'(z) = g(z)h^{\beta}(z) \in P$$ , $z \in E$ . Now, for $j \ge 1$ , $z_1$ any non-zero complex number and $z = re^{i\theta}$ , we consider for F(z) = zf'(z), $$\left| \Delta_j(n, z_1, F) \right|$$ $$= \left| \frac{1}{2\pi r^{n+j}} \int_0^{2\pi} (z - z_1)^j F(z) e^{-i(n+j)\theta} d\theta \right|$$ <span id="page-8-0"></span> $$\leq \frac{1}{2\pi r^{n+j}} \int_{0}^{2\pi} |z-z_{1}|^{j} \left| z^{1-(1-\rho)\eta} \frac{(s_{1}(z)/z)^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\eta}}{(s_{2}(z)/z)^{(\frac{k}{4}-\frac{1}{2})(1-\rho)\eta}} \right| \left| h^{\beta}(z) \right| d\theta$$ $$\leq \frac{1}{2\pi r^{n+j}} \int_{0}^{2\pi} |z-z_{1}|^{j} \left| s_{1}(z) \right|^{j} \frac{|s_{1}(z)|^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-j}}{|s_{2}(z)|^{(\frac{k}{4}-\frac{1}{2})(1-\rho)\operatorname{Re}\eta}} \left| h(z) \right|^{\beta} d\theta,$$ where we have used Lemma 1.1(iii). Using Lemma 1.3, we have $$\left|\Delta_{j}(n,z_{1},F)\right| \leq \frac{1}{2\pi r^{n+j}} \left(\frac{2r^{2}}{1-r^{2}}\right)^{j} \int_{0}^{2\pi} \frac{|s_{1}(z)|^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-j}}{|s_{2}(z)|^{(\frac{k}{4}-\frac{1}{2})(1-\rho)\operatorname{Re}\eta}} \left|h(z)\right|^{\beta} d\theta. \tag{3.3}$$ By employing distortion results for starlike functions and simplifying, we obtain, from (3.3), $$\left|\Delta_{j}(n,z_{1},F)\right| \leq \frac{1}{2\pi} \frac{(2)^{(\frac{k}{2}-1)(1-\rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4}-\frac{1}{2})(1-\rho)\operatorname{Re}\eta+n-j-1}} \left(\frac{1}{1-r}\right)^{j} \int_{0}^{2\pi} \left|s_{1}(z)\right|^{(\frac{k}{4}+\frac{1}{2})(1-\rho)\operatorname{Re}\eta-j} \left|h(z)\right|^{\beta} d\theta.$$ Using Holder's inequality, with $p = \frac{2}{2-\beta}$ , $q = \frac{2}{\beta}$ , such that $\frac{1}{p} + \frac{1}{q} = 1$ , we can write $$\begin{split} \left| \Delta_{j}(n, z_{1}, F) \right| &\leq \frac{(2)^{(\frac{k}{2} - 1)(1 - \rho)\operatorname{Re}\eta}}{r^{(\frac{k}{4} - \frac{1}{2})(1 - \rho)\operatorname{Re}\eta + n - j - 1}} \left( \frac{1}{1 - r} \right)^{j} \left( \frac{1}{2\pi} \int_{0}^{2\pi} \left| s_{1}(z) \right|^{\frac{(\frac{k}{2} + 1)(1 - \rho)\operatorname{Re}\eta - 2j}{2 - \beta}} \right)^{\frac{2 - \beta}{2}} \\ &\times \left( \frac{1}{2\pi} \int_{0}^{2\pi} \left| h(z) \right|^{2} d\theta \right)^{\frac{\beta}{2}}. \end{split}$$ Proceeding in a similar way as in Theorem 2.3, we have $$\begin{split} \left| \Delta_{j}(n, z_{1}, F) \right| &\leq \frac{(2)^{(\frac{k}{2} - 1)(1 - \rho)\operatorname{Re}\eta + \frac{\beta}{2}}}{r^{n - 1}} \left( \frac{1}{1 - r} \right)^{\frac{\beta}{2} + j} \\ &\times \left( \frac{1}{2\pi} \int_{0}^{2\pi} \frac{1}{\left| 1 - re^{i\theta} \right|^{\frac{(k + 2)(1 - \rho)\operatorname{Re}\eta - 4j}{2 - \beta}}} d\theta \right)^{\frac{2 - \beta}{2}}. \end{split}$$ Subordination for starlike functions further yields $$|\Delta_j(n, z_1, F)| = O(1) \left(\frac{1}{1-r}\right)^{(\frac{k}{2}+1)(1-\rho)\operatorname{Re} \eta - j + \beta - 1},$$ where O(1) depends only on k, $\eta$ , $\beta$ and j. Now, applying Lemma 3.2 and putting $z_1 = \frac{n}{n+1}e^{i\theta_m}$ $(n \to \infty)$ , we have for $k \ge (\frac{4j}{(1-\rho)\operatorname{Re}\eta} - 2)$ , $j \ge 1$ , $$\Delta_j(n, e^{i\theta_n}, f(z)) = O(1)n^{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta - j + \beta - 1}.$$ We now estimate the growth rate of $H_q(n)$ . For q = 1, $H_q(n) = a_n = \Delta_0(n)$ and from Theorem 2.4, it follows that $$H_1(n) = O(1)n^{(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta + \beta - 2}$$ For q ≥ , we use Remark [.](#page-7-0) together with Lemma [.](#page-6-0), to have $$H_q(n) = \mathrm{O}(1) n^{q[(\frac{k}{2}+1)(1-\rho)\operatorname{Re}\eta + \beta] - q^2}, \quad k > \left(\frac{4(q-1)}{(1-\rho)\operatorname{Re}\eta} - 2\right),$$ where O() depends only on k, η, ρ, β and q. - By giving special values to the parameters involved in the above theorem, we obtain the following interesting results.

Corollary . Let f ^∈ Tk(ρ,β) and be defined as in ([.\)](#page-0-1). Then, for q ^≥ , ^k > ( (q–) –^ρ – ), $$H_q(n) = O(1)n^{q[(\frac{k}{2}+1)(1-\rho)+\beta]-q^2} \quad (n \to \infty),$$ where O() depends only on k, ρ, β and q. Noor [[\]](#page-9-5) studied the above corollary with a different method.

Corollary . Let f ∈ Tk and be defined as in ([.](#page-0-1)). Then, for q ≥ , k > (q – ), <span id="page-9-1"></span> $$H_q(n) = O(1)n^{q[(\frac{k}{2}+2]-q^2} \quad (n \to \infty),$$ where O() depends only on k and q.

Definition 1.1. Let $f \in A$ be locally univalent in E. Then, for $\eta \neq 0$ (complex), $0 \leq \rho < 1$ , $f \in V_k(\eta, \rho)$ if and only if $$\left(1 + \frac{1}{\eta} \frac{zf''(z)}{f'(z)}\right) \in P_k(\rho), \quad z \in E.$$ We note that for $\eta=1$ , we have the class $V_k(\rho)$ of bounded boundary rotations of order $\rho$ introduced by Padmanabhan and Parvatham [1]. Also, $V_k(0)=V_k$ , the class of functions of bounded boundary rotations and $V_2(\eta,0)=C(\eta)$ denotes the class of convex functions of complex order.

Definition 1.2. Let $f \in A$ . Then $f \in \widetilde{N}_k(\eta, \rho, \beta)$ if and only if, for $k \ge 2$ , $\beta \ge 0$ , there exists a function $g \in V_k(\eta, \rho)$ such that $$\left|\arg\frac{f'(z)}{g'(z)}\right| \le \frac{\beta\pi}{2}, \quad z \in E.$$ <span id="page-1-0"></span>For $\eta=1$ , we have the class $\widetilde{T}_k(\rho,\beta)$ which was recently introduced and studied by Noor [4]. For k=2, $\eta=1$ , $\rho=0$ , $\widetilde{N}_2(1,0,\beta)$ is the class of strongly close-to-convex functions. Also, $\widetilde{N}_2(1,\rho,0)=C(\rho)$ is the class of convex functions of order $\rho$ . For $\eta=1$ , $\rho=0$ , $\beta=1$ , the class of $\widetilde{N}_k(\eta,\rho,\beta)$ reduces to the class $T_k$ introduced by Noor [3]. We need the following results in our investigation.

Definition 3.1. Let $z_1$ be a non-zero complex number. Then for f(z), given by (1.1), we define $$\Delta_i(n, z_1, f(z)) = \Delta_{i-1}(n, z_1, f(z)) - z_1 \Delta_{i-1}(n+1, z_1, f(z)), \quad j > 1$$ <span id="page-6-0"></span>with $\Delta_1(n, z_1, f(z)) = a_n$ . The following two lemmas are due to Noonan and Thomas [10] which are essential in our investigations.