In this article, we introduce new subclasses of normalized analytic functions in the unit disk \(U\), defined by a generalized Raducanu-Orhan differential Operator. Various results are driven including coefficient inequalities, growth and distortion theorem, closure property, \(\delta$\)-neighborhoods, extreme points, radii of close-to-convexity, starlikeness and convexity for these subclasses.
Remark 1.
Remark 2.
Definition 1. A function \(f\in \mathcal{T}_{p}\) is in the class, \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if
Remark 3. If \(\nu=0\), \(\vartheta=\alpha\), \(\varphi=\mu\), the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) reduces to the class \(R_{p}^n(\alpha,\beta,\gamma,\mu)\) investigated by Bulut [4]
Definition 2. A function \(f\in \mathcal{T}_{p}\) is in the class \(S_{p}^{n,(\delta_0)}(\vartheta,\beta,\gamma,\varphi)\), if there exists a function \(g(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) such that $$\left|\frac{f(z)}{g(z)}-1\right|< 1-\delta_0 … (z\in U ,0\le \delta_0 < 1)$$ for \(0\le \vartheta< 1,\) \(0\le\gamma< 1,\) \(0< \beta\le1,\) \(0< \varphi< 1.\)
Definition 3. For a function \(f\in \mathcal{T}_{p}\), \(\delta\ge 0\), \(\delta\)-neighborhood of \(f\) is defined as:
Theorem 4. A function \(f\in \mathcal{T}_{p}\) is in the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if
Proof. Suppose that \(f\in S_{p}^n(\vartheta, \beta, \gamma, \varphi),\) then from inequality 10, we have \begin{eqnarray*} \left| \frac {(D_{\alpha \nu}^{n,p}f(z))’ -pz^{p-1}}{\vartheta(D_{\alpha \nu}^{n,p}f(z))’+(\beta-\gamma)} \right|&=&\left| \frac {pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1} -pz^{p-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)} \right|\\ &= &\left| \frac {\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)} \right|\\&<&\varphi, (z\in U,n\in N_{0}) \end{eqnarray*} it is well known that \(\Re z\le \left|z\right|\), therefore, we obtain \begin{align*} \Re\left\{\frac {\sum_{K=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1}}{\vartheta(pz^{p-1}-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}z^{k-1})+(\beta-\gamma)}\right\} < \varphi. \end{align*} If we choose \(z\) real and let \(z \rightarrow 1^-,\) then we get \begin{eqnarray*} \sum_{K=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}&\le& \varphi\{\vartheta(p-\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k})+(\beta-\gamma)\} \end{eqnarray*} which is precisely the assertion 13. On contrary, suppose that the inequality 13 hold true and let \(z\in \delta U=\lbrace z\in C:\left|z\right|=1\rbrace.\) Then, from 10, we have \begin{eqnarray*} &&\left|(D_{\alpha \nu}^{n,p}f(z))' -pz^{p-1} \right|- \varphi\left|\vartheta(D_{\alpha \nu}^{n,p}f(z))'+(\beta-\gamma) \right|\le \sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}\\&&- \varphi(\vartheta p+\beta-\gamma)+ \varphi \vartheta \sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}\\ &&=\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}a_{k}\left| z\right|^{k-1}(1+\varphi\vartheta)a_{k}- \varphi(\vartheta p+\beta-\gamma)\le 0. \end{eqnarray*} By maximum modulus theorem, we have \(f \in S_{p}^n(\vartheta,\beta,\gamma,\varphi).\)
Corollary 5. If \(f \in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), then \( a_{p+1}\le \frac{\varphi(\vartheta p+\beta-\gamma)}{(p+1)\left[1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi\vartheta)}.\)
Theorem 6. For each \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have \( \left|z\right|^p-\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)(p+1)}\left|z\right|^{p+1} \le\left|f(z)\right| \le\left|z\right|^p+\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)(p+1)}\left|z\right|^{p+1}.\)
Proof. Let \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi),z\in U\), the bound on \(f(z)\) is given by
Theorem 7. For each \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have \( p\left|z\right|^{p-1}-\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)}\left|z\right|^{p} \le\left|f'(z)\right| \le p\left|z\right|^{p-1}+\frac{\varphi(\vartheta p+\beta-\gamma)}{\left[{1+(\alpha \nu (p+1)+\alpha-\nu)(\frac{1}{p})}\right]^{n}(1+\varphi \vartheta)}\left|z\right|^{p}.\)
Proof. Let \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi),z\in U\), the bound on the derivative of \(f(z)\) is given by \begin{align*} \left|f'(z)\right|\le p\left|z\right|^{p-1}+ (p+1)\left|z \right|^p\sum_{k=p+1}^\infty a_k, z\in U, \end{align*} and, in the same way as above, we get our desired result.
Theorem 8. Let the functions \begin{align*} f(z)=z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k} , && (a_k\ge 0)\\ g(z)=z^{p}-\sum_{k=p+1}^{\infty}b_{k}z^{k} , && (b_k\ge 0), \end{align*} be in the class \( S_{p}^n(\vartheta,\beta,\gamma,\varphi)\). Then for \(0\le \lambda\le1,\) the function \(h\) is defined as $$h(z)=(1-\lambda)f(z)+\lambda g(z)=z^{p}-\sum_{k=p+1}^{\infty}c_{k}z^{k},$$ where \( c_k:=(1-\lambda)a_k+\lambda b_k\ge0,\) is also in \( S_{p}^n(\vartheta,\beta,\gamma,\varphi).\)
Proof. Suppose that each of the functions \(f\) and \(g\) is in the class \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\). Then making use of inequality (13), we have \begin{eqnarray*} &&\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)c_{k}\\&&=(1-\lambda)\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k} \\&&+\lambda\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)b_{k} \\&& \le (1-\lambda)\varphi(\vartheta p+\beta-\gamma)+\lambda \varphi(\vartheta p+\beta-\gamma)\\&&= \varphi(\vartheta p+\beta-\gamma),\end{eqnarray*} which completes the proof.
Theorem 9. If
Proof. For a function \(f(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) of the form (8), Theorem 4 immediately yields $$ (p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)\sum_{k=p+1}^{\infty}a_{k}\le \varphi(\vartheta p+\beta-\gamma),$$ therefore,
Theorem 10. If \(g(z)\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) and
Proof. Suppose that \(f\in N_{\delta}^p(f,g)\), then by Definition 3, we have $$\sum_{k=p+1}^{\infty}k|a_k-b_k|\le \delta,$$ which readily implies the coefficient inequality given by $$\sum_{k=p+1}^{\infty}|a_k-b_k|\le \frac{\delta}{p+1} (p\in N).$$ Next, since \(g\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), we have from inequality (13) that \begin{align*} \sum_{k=p+1}^{\infty}b_{k}\le \frac{\varphi(\vartheta p+\beta-\gamma)}{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)}, \end{align*} so from the definition of the class, we have \begin{eqnarray*} \left|\frac{f(z)}{g(z)}-1\right|&<&\frac{\sum_{k=p+1}^{\infty}|a_k-b_k|}{1-\sum_{k=p+1}^\infty b_k}\\ &\le& \frac{\delta}{p+1}\frac{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)}{(p+1)\left[1+(\alpha \nu(p+1)+\alpha-\nu)(\frac{1}{p})\right]^{n}(1+\varphi \vartheta)-\varphi(\vartheta p+\beta-\gamma)}\\ &= &1-\delta_0,\end{eqnarray*} provided that \(\delta_0\) is given precisely by (22). Thus, by the definition, \(f\in S_{p}^{n,\delta_0}(\vartheta,\beta,\gamma,\varphi)\) for \(\delta_0\) given by (22), this completes our proof.
Theorem 11. If \( f_{p}(z)=z^{p} , f_{k}(z)=z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k} (k\ge p+1) \) then, \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) if and only if it can be expressed in the form \(f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z),\) where \(\lambda_k\ge 0\) and \(\lambda_{p}\)=\(1 – \sum_{k=p+1}^{\infty}\lambda_{k}\).
Proof. Assume that \(f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z)\), then \begin{eqnarray*} f(z)&=&(1 – \sum_{k=p+1}^{\infty}\lambda_{k})z^{p}+\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k}\right\}\\&=&z^{p}-\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k}\right\}.\end{eqnarray*} Thus, \begin{eqnarray*}&&\sum_{k=p+1}^{\infty}k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta) \lambda_{k} \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}\\&&= \varphi(\vartheta p+\beta-\gamma)\sum_{k=p+1}^{\infty}\lambda_{k}= \varphi(\vartheta p+\beta-\gamma)(1-\lambda_{p}) \le \varphi(\vartheta p+\beta-\gamma),\end{eqnarray*} which shows that \(f\) satisfies condition (13) and therefore,\(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi).\) Conversely, suppose that \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), since \[ a_{k} \le \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}, (k\ge p+1),\] we may set \[ \lambda_{k}=\frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}{\varphi(\vartheta p+\beta-\gamma)}a_{k}, \text{ and } \lambda_{p}=1- \sum_{k=p+1}^{\infty}\lambda_{k},\] then we obtain from \begin{eqnarray*}f(z)&=& z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}\\&=&(\lambda_{p}+\sum_{k=p+1}^{\infty}\lambda_{k})z^{p} – \sum_{k=p+1}^{\infty}\lambda_{k} \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^k\\ &=&\lambda_{p}z^{p}+ \sum_{k=p+1}^{\infty}\lambda_{k}(z^{p}- \frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^k)\\&=& \lambda_{p}z^{p} + \sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z),\end{eqnarray*} which completes the proof.
Corollary 12. The extreme points of \(S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) are given by $$f_{p}(z)=z^{p} , f_{k}(z)=z^{p}-\frac{\varphi(\vartheta p+\beta-\gamma)}{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)}z^{k} (k\ge p+1)$$
\rho,$$ for some \(\rho (0\le\rho
\rho,$$ for some \(\rho (0\le\rho< p)\) and for all \(z\in U.\)
Theorem 13. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) then \(f\) is \(p\)-valently close-to-convex of order \(\rho\) in \(\left|z\right|< r_1(\vartheta,\beta,\gamma,\varphi,\rho)\), where \begin{align*} r_1(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)a_{k}(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}
Proof. It is sufficient to show that \(\left|\frac{f'(z)}{z^{p-1}}-p\right|< p-\rho.\) Since \(\left|\frac{pz^{p-1}-\sum_{k=p+1}^\infty ka_Kz^{k-1}}{z^{p-1}}-p\right|< p-\rho,\) which implies that $$\left|\frac{f'(z)}{z^{p-1}}-p\right|\le \sum_{k=p+1}^{\infty}ka_k\left|z\right|^{k-p}< p-\rho,$$ implies
Theorem 14. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\) then \(f\) is \(p\)-valently starlike of order \(\rho\) in \(\left|z\right|< r_2(\vartheta,\beta,\gamma,\varphi,\rho)\), where \begin{align*} r_2(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{k\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)(k-\rho)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}
Proof. In order to prove, it suffices to show that \(\left|\frac{zf'(z)}{f(z)}-p\right| < p-\rho. \)
Theorem 15. If \(f\in S_{p}^n(\vartheta,\beta,\gamma,\varphi)\), then \(f\) is \(p\)-valently convex of order \(\rho\) in \(\left|z\right|< r_3(\vartheta, \beta, \gamma, \varphi, \rho)\), where \begin{align*} r_3(\vartheta,\beta,\gamma,\varphi,\rho)=\inf_{k}\left\{\frac{\left[{1+(\alpha \nu k+\alpha-\nu)\left(\frac{k}{p}-1\right)}\right]^{n}(1+\varphi \vartheta)p(p-\rho)}{\varphi(\vartheta p+\beta-\gamma)(k-\rho)} \right\}^{\frac{1}{k-p}} k\ge p+1. \end{align*}
Proof. To prove this, it suffices to show that \(\left|1+\frac{zf”(z)}{f'(z)}-p\right| < p-\rho .\) Since