Let $\Sigma$ denote the class of functions $f$ of the form

$\begin{equation}f(z)=\frac{1}{z}+\sum\limits_{n=1}^{\infty}a_{n}z^{n},\end{equation}$

(1.1)

which are analytic in the punctured disk $U^*=\{z:0<|z|<1\}$.

Also let $\Sigma_\alpha$ denote the class of functions of the form

$\begin{equation}F(z)=\frac{1}{z}+\sum\limits_{n=1}^{\infty}a_{n}z^{n-\frac{n}{\alpha}}(\alpha\in N\setminus\{1\}),\end{equation}$

(1.2)

which are analytic in the punctured disk $U^*$(cf. [1, 2]). When $\alpha$ goes to infinity then $(n-\frac{n}{\alpha})$ approaches $n$; hence $\Sigma_\alpha=\Sigma$.

Throughout this paper, let the functions of the form

$\begin{eqnarray}&&f(z)=\frac{a_0}{z}+\sum\limits_{n=1}^{\infty}a_{n}z^{n-\frac{n}{\alpha}}(a_0>0,a_n\geq0,\alpha\in N\setminus\{1\}),\end{eqnarray}$

(1.3)

$\begin{eqnarray} &&f_i(z)=\frac{a_{0,i}}{z}+\sum\limits_{n=1}^{\infty}a_{n,i}z^{n-\frac{n}{\alpha}}(a_{0,i}>0,a_{n,i}\geq0,\alpha\in N\setminus\{1\}),\end{eqnarray}$

(1.4)

$\begin{eqnarray} &&g(z)=\frac{b_0}{z}+\sum\limits_{n=1}^{\infty}b_{n}z^{n-\frac{n}{\alpha}}(b_0>0,b_n\geq0,\alpha\in N\setminus\{1\}),\end{eqnarray}$

(1.5)

and

$g_j(z)=\frac{b_{0,j}}{z}+\sum\limits_{n=1}^{\infty}b_{n,j}z^{n-\frac{n}{\alpha}}(b_{0,j}>0,b_{n,j}\geq0,\alpha\in N\setminus\{1\})$

(1.6)

be regular and univalent in the punctured disk $U^*$.

For the function $F\in\Sigma_\alpha$, we define

$\begin{eqnarray*}&&I_\alpha^0F(z)=F(z),\\ &&I_\alpha^1F(z)=zF'(z)+\frac{2}{z},\\ &&I_\alpha^2F(z)=z(I_\alpha^1F(z))'+\frac{2}{z},\end{eqnarray*}$

and for $k=1,2,\cdots$, we can write

$I_\alpha^kF(z)=z(I_\alpha^{k-1}F(z))'+\frac{2}{z} =\frac{1}{z}+\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^ka_{n}z^{n-\frac{n}{\alpha}},$

where $\alpha\in N\setminus\{1\},k\geq0$ and $z\in U^*$. We note that when $\alpha$ goes to $\infty$ then $n(\frac{\alpha-1}{\alpha})$ approaches $n$; in this way we have $I_\alpha^k\rightarrow I^k$, which was introduced by Frasin and Darus [3] (see also [4]).

With the help of the differential operator $I_\alpha^k$, we define the following subclasses of $\Sigma_\alpha$.

Let $\Sigma_\alpha S^*(k,\beta,\gamma)$ be the class of functions $F$ defined by (1.2) and satisfying the condition

$\left|\frac{z(I_\alpha^{k}F(z))'}{I_\alpha^{k}F(z)}+1\right| <\beta\left|\frac{z(I_\alpha^{k}F(z))'}{I_\alpha^{k}F(z)}+2\gamma-1\right|$

(1.7)

$(z\in U^*,0\leq\gamma<1,0<\beta\leq1,k\in N_0=N\cup\{0\},\alpha\in N\setminus\{1\}).$

Also let $\Sigma_\alpha C(k,\beta,\gamma)$ be the class of functions $F$ for which $-zF'(z)\in\Sigma_\alpha S^*(k,\beta,\gamma)$.

Using similar methods as given in [4], we can easily obtain the characterization properties for the classes $\Sigma_\alpha S^*(k,\beta,\gamma)$ and $\Sigma_\alpha C(k,\beta,\gamma)$ as follows.

Lemma 1.1 A function $f$ defined by (1.3) belongs to the class $\Sigma_\alpha S^*(k,\beta,\gamma)$, if and only if

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^k \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n}\leq2\beta(1-\gamma)a_0.$

(1.8)

Lemma 1.2 A function $f$ defined by (1.3) belongs to the class $\Sigma_\alpha C(k,\beta,\gamma)$ if and only if

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{k+1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n}\leq2\beta(1-\gamma)a_0.$

(1.9)

We also note that when $\alpha$ goes to $\infty$ then we have $\Sigma_\alpha S^*(k,\beta,\gamma)\rightarrow \Sigma S^*(k,\beta,\gamma)$ and $\Sigma_\alpha C(k,\beta,\gamma)\rightarrow \Sigma C(k,\beta,\gamma)$, which are special classes that were introduced by El-Ashwah and Aouf [4].

Now, we introduce the following class of meromorphic univalent functions in $U^*$.

Definition 1.1 A function $f$ of form (1.3), which is analytic in $U^*$, belongs to the class $\Sigma_\alpha^h(\beta,\gamma)$ if and only if

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n}\leq2\beta(1-\gamma)a_0,$

(1.10)

where $0\leq\gamma<1,0<\beta\leq1,\alpha\in N\setminus\{1\}$ and $h$ is any fixed nonnegative real number. The class $\Sigma_\alpha^h(\beta,\gamma)$ is nonempty for any nonnegative real number $h$ as the functions have the form

$f(z)=\frac{a_0}{z}+\sum\limits_{n=1}^{\infty}\frac{2\beta(1-\gamma)a_0} {\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]} \lambda_{n}z^{n-\frac{n}{\alpha}},$

(1.11)

where $a_0>0$, $\lambda_n\geq0$ and $\sum\limits_{n=1}^{\infty}\lambda_n\leq1$, satisfying inequality (1.10).

Clearly, we have the following relationships:

(ⅰ) $\Sigma_\alpha^k(\beta,\gamma)\equiv\Sigma_\alpha S^*(k,\beta,\gamma)$and$\Sigma_\alpha^{k+1}(\beta,\gamma)\equiv\Sigma_\alpha C(k,\beta,\gamma)$;

(ⅱ) $\Sigma_\alpha^{h_1}(\beta,\gamma)\subset\Sigma_\alpha^{h_2}(\beta,\gamma)(h_1>h_2\geq0)$;

(ⅲ) $\Sigma_\alpha^h(\beta,\gamma)\subset\Sigma_\alpha^{h-1}(\beta,\gamma)\subset\cdots\subset\Sigma_\alpha C(k,\beta,\gamma)\subset\Sigma_\alpha S^*(k,\beta,\gamma)(h>k+1)$.

Following the earlier works of Mogra [5, 6] and Aouf and Darwish [7] (see also [4, 8]), we define the quasi-Hadamard product of the functions $f(z)$ and $g(z)$ by

$f*g(z)=\frac{a_0b_0}{z}+\sum\limits_{n=1}^{\infty}a_{n}b_{n}z^{n-\frac{n}{\alpha}}.$

(1.12)

Similarly, we can define the quasi-Hadamard product of more than two functions, e.g.,

$f_1*f_2*\cdots*f_p(z)=\left(\prod\limits_{i=1}^{p}a_{0,i}\right)z^{-1}+\sum\limits_{n=1}^{\infty} \left(\prod\limits_{i=1}^{p}a_{n,i}\right)z^{n-\frac{n}{\alpha}},$

(1.13)

where the functions $f_i(i=1,2,\cdots,p)$ are given by (1.4).

The object of this paper is to derive certain results related to the quasi-Hadamard product of functions belonging to the classes $\Sigma_\alpha^h(\beta,\gamma)$, $\Sigma_\alpha S^*(k,\beta,\gamma)$ and $\Sigma_\alpha C(k,\beta,\gamma)$.

Unless otherwise mentioned, we shall assume throughout the following results that $z\in U^*,0\leq\gamma<1,0<\beta\leq1,k\in N_0,\alpha\in N\setminus\{1\}$ and $h$ is any fixed nonnegative real number.

Theorem 2.1 Let the functions $f_i(z)$ defined by (1.4) be in the class $\Sigma_\alpha C(k,\beta,\gamma)$ for every $i=1,2,\cdots,p$; and let the functions $g_j(z)$ defined by (1.6) be in the class $\Sigma_\alpha^h(\beta,\gamma)$ for every $j=1,2,\cdots,q$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{p(k+2)+q(h+1)-1}(\beta,\gamma)$.

Proof Let $G(z)=f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$, then

$\begin{equation}G(z)=\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right)z^{-1}+\sum\limits_{n=1}^{\infty} \left(\prod\limits_{i=1}^{p}a_{n,i}\prod\limits_{j=1}^{q}b_{n,j}\right)z^{n-\frac{n}{\alpha}}.\end{equation}$

(2.1)

It is sufficient to show that

$\begin{eqnarray}&&\sum\limits_{n=1}^{\infty}\left\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(k+2)+q(h+1)-1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]\right.\nonumber\\ &&\left. \left(\prod\limits_{i=1}^{p}a_{n,i}\prod\limits_{j=1}^{q}b_{n,j}\right)\right\} \leq2\beta(1-\gamma)\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right).\end{eqnarray}$

(2.2)

Since $f_i\in \Sigma_\alpha C(k,\beta,\gamma)$, by Lemma 1.2 we have

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{k+1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n,i}\leq2\beta(1-\gamma)a_{0,i}$

(2.3)

for every $i=1,2,\cdots,p$. Thus,

$\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{k+1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n,i}\leq2\beta(1-\gamma)a_{0,i}$

or

$a_{n,i}\leq\frac{2\beta(1-\gamma)}{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{k+1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]}a_{0,i}$

for every $i=1,2,\cdots,p$. The right-hand expression of the last inequality is not greater than $\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(k+2)}a_{0,i}$. Therefore,

$a_{n,i}\leq\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(k+2)}a_{0,i}$

(2.4)

for every $i=1,2,\cdots,p$. Also, since $g_j\in\Sigma_\alpha^h(\beta,\gamma)$, we find from (1.10) that

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]b_{n,j}\leq2\beta(1-\gamma)b_{0,j},$

(2.5)

which implies that

$b_{n,j}\leq\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(h+1)}b_{0,j}$

(2.6)

for every $j=1,2,\cdots,q$.

Using (2.4)-(2.6) for $i=1,2,\cdots,p; j=q$; and $j=1,2,\cdots,q-1$ respectively, we have

$\begin{eqnarray*}&&\sum\limits_{n=1}^{\infty}\left\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(k+2)+q(h+1)-1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right] \left(\prod\limits_{i=1}^{p}a_{n,i}\prod\limits_{j=1}^{q}b_{n,j}\right)\right\}\\ &\leq&\sum\limits_{n=1}^{\infty}\biggl\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(k+2)+q(h+1)-1} \cdot \left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-p(k+2)}\cdot \left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(q-1)(h+1)}\\ &&\cdot\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q-1}b_{0,j}\right) \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]b_{n,q}\biggr\}\end{eqnarray*}$

$\begin{eqnarray*}&\leq&\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q-1}b_{0,j}\right) \left\{\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]b_{n,q}\right\} \\ &\leq&2\beta(1-\gamma)\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right).\end{eqnarray*}$

Thus, we have $G(z)\in\Sigma_\alpha^{p(k+2)+q(h+1)-1}(\beta,\gamma)$. This completes the proof of Theorem 2.1.

Upon setting $h=k+1$ in Theorem 2.1, we obtain the following result.

Corollary 2.1 Let the functions $f_i(z)$ defined by (1.4) and the functions $g_j(z)$ defined by (1.6) belong to the class $\Sigma_\alpha C(k,\beta,\gamma)$ for every $i=1,2,\cdots,p$ and $j=1,2,\cdots,q$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{p(k+2)+q(k+2)-1}(\beta,\gamma)$.

Theorem 2.2 Let the functions $f_i(z)$ defined by (1.4) be in the class $\Sigma_\alpha^h(\beta,\gamma)$ for every $i=1,2,\cdots,p$; and let the functions $g_j(z)$ defined by (1.6) be in the class $\Sigma_\alpha S^*(k,\beta,\gamma)$ for every $j=1,2,\cdots,q$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{p(h+1)+q(k+1)-1}(\beta,\gamma)$.

Proof Suppose that $G(z)$ be defined as (2.1). To prove the theorem, we need to show that

$\sum\limits_{n=1}^{\infty}\left\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(h+1)+q(k+1)-1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right] \left(\prod\limits_{i=1}^{p}a_{n,i}\prod\limits_{j=1}^{q}b_{n,j}\right)\right\} $

$\leq2\beta(1-\gamma)\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right).$

(2.7)

Since $f_i\in\Sigma_\alpha^h(\beta,\gamma)$, from (1.10) we have

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n,i}\leq2\beta(1-\gamma)a_{0,i},$

(2.8)

which implies that

$a_{n,i}\leq\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(h+1)}a_{0,i}$

(2.9)

for every $i=1,2,\cdots,p$. Further, since $g_j\in\Sigma_\alpha S^*(k,\beta,\gamma)$, by Lemma 1.1 we have

$\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^k \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]b_{n,j}\leq2\beta(1-\gamma)b_{0,j}$

for every $j=1,2,\cdots,q$. Whence we obtain

$b_{n,j}\leq\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(k+1)}b_{0,j}$

(2.10)

for every $j=1,2,\cdots,q$.

Using (2.8)-(2.10) for $i=p; i=1,2,\cdots,p-1$; and $j=1,2,\cdots,q$ respectively, we get

$\begin{eqnarray*}&&\sum\limits_{n=1}^{\infty}\left\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(h+1)+q(k+1)-1} \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right] \left(\prod\limits_{i=1}^{p}a_{n,i}\prod\limits_{j=1}^{q}b_{n,j}\right)\right\}\\ &\leq&\sum\limits_{n=1}^{\infty}\biggl\{\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{p(h+1)+q(k+1)-1} \cdot\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-(p-1)(h+1)}\cdot \left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^{-q(k+1)}\\ &&\cdot\left(\prod\limits_{i=1}^{p-1}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right) \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n,p}\biggr\}\\ &\leq&\left(\prod\limits_{i=1}^{p-1}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right) \left\{\sum\limits_{n=1}^{\infty}\left[n\left(\frac{\alpha-1}{\alpha}\right)\right]^h \left[n\left(\frac{\alpha-1}{\alpha}\right)(1+\beta)+(2\gamma-1)\beta+1\right]a_{n,p}\right\}\\ &\leq&2\beta(1-\gamma)\left(\prod\limits_{i=1}^{p}a_{0,i}\prod\limits_{j=1}^{q}b_{0,j}\right).\end{eqnarray*}$

Therefore, we have $G(z)\in\Sigma_\alpha^{p(h+1)+q(k+1)-1}(\beta,\gamma)$. We complete the proof.

By taking $h=k$ in Theorem 2.2, we get the following result.

Corollary 2.2 Let the functions $f_i(z)$ defined by (1.4) and the functions $g_j(z)$ defined by (1.6) belong to the class $\Sigma_\alpha S^*(k,\beta,\gamma)$ for every $i=1,2,\cdots, p$ and $j=1,2,\cdots, q$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{p(k+1)+q(k+1)-1}(\beta,\gamma)$.

By putting $h=k$ in Theorem 2.1 or $h=k+1$ in Theorem 2.2, we obtain the following result.

Corollary 2.3 Let the functions $f_i(z)$ defined by (1.4) be in the class $\Sigma_\alpha C(k,\beta,\gamma)$ for every $i=1,2,\cdots,p$; and let the functions $g_j(z)$ defined by (1.6) be in the class $\Sigma_\alpha S^*(k,\beta,\gamma)$ for every $j=1,2,\cdots,q$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p*g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{p(k+2)+q(k+1)-1}(\beta,\gamma)$.

Next, we discuss some applications of Theorems 2.1 and 2.2.

Taking into account the quasi-Hadamard product of functions $f_1(z),f_2(z),\cdots,f_p(z)$ only, in the proof of Theorem 2.1, and using (2.3) and (2.4) for $i=p$ and $i=1,2,\cdots, p-1$, respectively, we are led to

Corollary 2.4 Let the functions $f_i(z)$ defined by (1.4) belong to the class $\Sigma_\alpha C(k,\beta,\gamma)$ for every $i=1,2,\cdots,p$. Then the quasi-Hadamard product $f_1*f_2*\cdots*f_p(z)$ belongs to the class $\Sigma_\alpha^{p(k+2)-1}(\beta,\gamma)$.

Also, taking into account the quasi-Hadamard product of functions $g_1(z),g_2(z),\cdots,g_q(z)$ only, in the proof of Theorem 2.2, and using (2.10) and (2.11) for $j=q$ and $j=1,2,\cdots, q-1$, respectively, we are led to

Corollary 2.5 Let the functions $g_j(z)$ defined by (1.6) belong to the class $\Sigma_\alpha S^*(k,\beta,\gamma)$ for every $j=1,2,\cdots,q$. Then the quasi-Hadamard product $g_1*g_2*\cdots*g_q(z)$ belongs to the class $\Sigma_\alpha^{q(k+1)-1}(\beta,\gamma)$.

Remark 2.1 By letting $\alpha\rightarrow\infty$ in the proofs of Corollaries 3-5, we obtain the results obtained by El-Ashwah and Aouf [4, Theorems 3, 1 and 2, respectively].

Remark 2.2 By letting $\alpha\rightarrow\infty$ and $k=0$ in the proofs of Corollaries 3-5, we obtain the results obtained by Mogra [6, Theorems 3, 1 and 2, respectively].