Abstract
For \(\alpha , \gamma \ge 0\) and \(\beta <1\), let \({\mathcal {W}}_{\beta }(\alpha , \gamma )\) denote the class of all normalized analytic functions \(f\) in the open unit disk \(E=\{z:|z|<1\}\) such that
for some \(\phi \in {\mathbb {R}}\). For \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\), we consider the integral transform
where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). In a very recent paper, Ali et al. (J Math Anal Appl 385:808–822, 2012) discussed the starlikeness of the integral transform \(V_{\lambda }(f)\) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). The aim of present paper is to find conditions on \(\lambda (t)\) such that \(V_{\lambda }(f)\) is starlike of order \(\delta \) (\(0\le \delta \le 1/2\)) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). As applications, we study various choices of \(\lambda (t)\), related to classical integral transforms.
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1 Introduction
Let \({\mathcal {A}}\) denotes the class of analytic functions \(f\) defined in the open unit disk \(E=\{z : |z|<1 \}\) with the normalization \(f(0)=f'(0)-1=0\) and \({\mathcal {A}}_{0}=\left\{ g: \, g(z)=f(z)/z, \, f\in {\mathcal {A}}\right\} \). Let \(S\) be the subclass of \(\mathcal {A}\) consisting of functions univalent in \(E\). A function \(f\) in \(\mathcal {A}\) is said to be starlike of order \(\beta \) if it satisfies
for some \(\beta \, \, (0\le \beta <1)\). We denote by \(S^{*}(\beta )\), the subclass of \(S\) consisting of functions which are starlike of order \(\beta \) in \(E\). Set \(S^{*}(0)=S^{*}\). For any two functions \(f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots \) and \(g(z)=z+b_{2}z^{2}+b_{3}z^{3}+\cdots \) in \(\mathcal {A}\), the Hadamard product (or convolution) of \(f\) and \(g\) is the function \(f *g\) defined by
For \(f\in {\mathcal {A}}\), Fournier and Ruscheweyh [4] introduced the operator
where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). This operator contains some of the well-known operators (such as Alexander, Libera, Bernardi, and Komatu) as its special cases. This operator has been studied by a number of authors for various choices of \(\lambda (t)\) [1, 3, 4, 6, 9, 12]. Fournier and Ruscheweyh [4] applied the Duality theory [10, 11] to prove the starlikeness of the linear integral transform \(V_{\lambda }(f)\) over functions \(f\) in the class
In 2001, Kim and Rønning [5] investigated starlikeness properties of the integral transform (1.1) for functions \(f\) in the class
Recently in 2008, Ponnusamy and Rønning [9] discussed the same problem for the functions in the class
It is evident that \({\mathcal {R}}_{\gamma }(\beta )\) is closely related to the class \({\mathcal {P}}_{\gamma }(\beta )\). Clearly, \(f\in {\mathcal {R}}_{\gamma }(\beta )\) if and only if \(zf'\) belongs to \({\mathcal {P}}_{\gamma }(\beta )\).
In a very recent paper, Ali et al. [1] discussed this problem for the functions \(f\) in the class
Note that \({{\mathcal {W}}_{\beta }(1, 0)}\equiv {\mathcal {P}}(\beta )\), \({{\mathcal {W}}_{\beta }(\alpha , 0)}\equiv {\mathcal {P}}_{\alpha }(\beta )\) and \({{\mathcal {W}}_{\beta }(1+2\gamma , \gamma )}\equiv {\mathcal {R}}_{\gamma }(\beta )\).
In Sect. 3 of the paper, we shall mainly tackle the problem: For given \(0\le \delta \le 1/2\), to find conditions on \(\beta \) such that \(V_{\lambda }(f)\in S^{*}(\delta )\) whenever \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\). In Sect. 4, we find easier criteria of starlikeness of the integral operator \(V_{\lambda }(f)\). While in the last section of the paper, we discussed applications of results obtained for various choices of \(\lambda (t)\).
To prove our result, we shall need the duality theory for convolutions, so we include here some basic concepts and results from this theory. For a subset \(\mathcal {B}\subset {\mathcal {A}}_{0}\) we define
The set \({\mathcal {B}}^{*}\) is called the dual of \({\mathcal {B}}\). Further, the second dual of \({\mathcal {B}}\) is defined as \({\mathcal {B}}^{**}=({\mathcal {B}}^{*})^{*}\). The basic reference to this theory is the book by Ruscheweyh [11] (see also [10]). We shall need the following fundamental result.
Theorem 1.1
(Duality Principle) Let
We have
-
(1)
\({\mathcal {B}}^{**}=\left\{ g\in {\mathcal {A}}_{0}: \, \exists \phi \in {{\mathbb {R}}} \, \text {such that} \, \mathfrak {R}\left( e^{i\phi }(g(z)-\beta )\right) >0, \, \, z\in E\right\} \).
-
(2)
If \(\Gamma _{1}\) and \(\Gamma _{2}\) are two continuous linear functionals on \(\mathcal {B}\) with \(0\not \in \Gamma _{2}\), then for every \(g\in {\mathcal {B}}^{**}\) we can find \({v}\in {\mathcal {B}}\) such that
$$\begin{aligned} \frac{\Gamma _{1}(g)}{\Gamma _{2}(g)}=\frac{\Gamma _{1}(v)}{\Gamma _{2}(v)}. \end{aligned}$$
2 Preliminaries
We use the notations introduced in [1]. Let \(\mu \ge 0\) and \(\nu \ge 0\) satisfy
For \(\gamma =0\), \(\mu \) is also taken to be 0, in which case, \(\nu =\alpha \ge 0\). Writing \(\alpha =1+2\gamma \) in (2.1), we get \(\mu +\nu =1+\gamma =1+\mu \nu \), or \((\mu -1)(1-\nu )=0\).
-
(i)
When \(\gamma >0\), then writing \(\mu =1\) gives \(\nu =\gamma \).
-
(ii)
If \(\gamma =0\), then \(\mu =0\) and \(\nu =\alpha =1\).
In the particular case \(\alpha =1+2\gamma \), the values of \(\mu \) and \(\nu \) for \(\gamma >0\) will be taken as \(\mu =1\) and \(\nu =\gamma \) respectively, while in the case when \(\gamma =0\), we have \(\mu =0\) and \(\nu =1=\alpha \).
Define
and
Here \({\phi }_{\mu ,\nu }^{-1}\) denotes the convolution inverse of \({\phi }_{\mu ,\nu }\) such that \({\phi }_{\mu ,\nu }*{\phi }_{\mu ,\nu }^{-1}=z/(1-z)\). If we take \(\gamma =0\), then \(\mu =0\), \(\nu =\alpha \) in (2.3), we have
If \(\gamma >0\), then \(\nu >0\), \(\mu >0\), and making the change of variables \(u=t^{\nu }\), \(v=s^{\mu }\) results in
Thus the function \({\psi }_{\mu ,\nu }\) can be written as
Further let \(g\) be the solution of the initial-value problem
satisfying \(g(0)=1\), where \(\delta \in [0,1/2]\). A simple calculation leads to the solution given by
In particular
3 Main Results
Theorem 3.1
Let \(\mu \ge 0\), \(\nu \ge 0\) satisfy (2.1), and \(\beta <1\) satisfy
where \(g\) is the solution of the initial-value problem (2.5). Further let
and assume that \(\displaystyle t^{1/{\nu }}\Lambda _{\nu }(t)\rightarrow 0\), and \(\displaystyle t^{1/{\mu }}\Pi _{\mu ,\nu }(t)\rightarrow 0\) as \(t\rightarrow 0^{+}\). Then for \(\delta \in [0,1/2]\), we have \(V_{\lambda }({{\mathcal {W}}}_{\beta }(\alpha , \gamma )) \subset S^{*}(\delta )\) if and only if \({\mathcal {L}}_{\Pi _{\mu ,\nu }}(h_{\delta })\ge 0\), where \({\mathcal {L}}_{\Pi _{\mu ,\nu }}(h_{\delta })\) and \(h_{\delta }\) are defined by following equations:
and
respectively. This conclusion does not hold for any smaller values of \(\beta \).
Proof
The case \(\gamma =0 (\mu =0,\, \nu =\alpha \)) corresponds to the Theorem 1.2 in [2], so we will prove the result only when \(\gamma >0\).
Let
Since \(\mu +\nu =\alpha -\gamma \) and \(\mu \nu =\gamma \), then
Writing \(f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}\), we obtain from (2.2)
and (2.3) gives that
Now, let \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\). Then, in the view of the Theorem 1.1, we may restrict our attention to functions \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\) for which
Thus (3.7) gives
and therefore
here \(\psi :=\psi _{\mu ,\nu }\). \(\square \)
Also, a well-known result from the theory of convolutions [11] (see also [10]) implies that
where
Hence \(F\in S^{*}(\delta )\) if and only if
Using (3.9), we have
This holds if and only if [10], p. 23]
which can also be written as
Writing \(w=tu\), we get
An integration by parts with respect to \(t\) and (2.5) gives
Again writing \(w=vt\) and \(\eta =st\) reduces the above inequality to
which after integration by parts with respect to \(t\) yields
Thus \(F\in S^{*}(\delta )\) if and only if \({\mathcal {L}}_{\Pi _{\mu ,\nu }}(h_{\delta })\ge 0\).
Finally, to prove the sharpness, let \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\) be of the form for which
Using a series expansion we obtain that
Thus
where \(\tau _{n}=\int _{0}^{1}\lambda (t)t^{n}\mathrm{d}t\). From (2.6), it is a simple exercise to write \(g(t)\) in a series expansion as
Now, by (3.1) and (3.10), we have
Therefore
Finally, we see that
For \(z=-1\), we have
Thus \(zF'(z)/F(z)\) at \(z=-1\) equals \(\delta \). This implies that the result is sharp for the order of starlikeness.
4 Consequences of Theorem 3.1
Theorem 4.1
Let \(0\le \delta \le 1/2\). Assume that both \(\Pi _{\mu ,\nu }(t)\) and \(\Lambda _{\nu }(t)\), as given in Theorem 3.1, are integrable on [0,1] and positive on (0,1). Further assume that \(\mu \ge 1\), and
If \(\beta \) satisfies (3.1), then we have \(V_{\lambda }({\mathcal {W}}_{\beta }(\alpha , \gamma ))\subset S^{*}(\delta )\), where \(V_{\lambda }(f)\) is defined by (1.1).
Proof
For \(\mu \ge 1\), the function \(t^{{1/{\mu }}-1}\) is decreasing on (0,1). Thus the condition (4.1) along with [8], Theorem 2.3] gives
The result now, follows from Theorem 3.1. \(\square \)
Below, we obtain the conditions to ensure starlikeness of \(V_{\lambda }(f)\). As defined in Theorem 3.1, for \(\gamma >0\),
In order to apply Theorem 4.1, we have to prove that the function
is decreasing in (0,1). Since \(p(t)>0\) and
or equivalently,
so it remains to show that \(q(t)\) is increasing over (0,1), where
Since \(q(1)=0\), this will imply that \(q(t)\le 0\), and thus \(p(t)\) is decreasing on (0,1). Now
So, \(q'(t)\ge 0\) for \(t\in (0,1)\) is equivalent to the inequality \(r(t)\le 0\), where \(r(t)\) is equal to
By using the idea similar to the one used to prove Theorem 3.1 in [3], we can write
where,
Clearly, \(A(t)>0\) and \(X(t)>0\) for all \(t\in (0,1).\)
Case (i). If \(Y(t)\le 0\) on (0,1), then \(r(t)\le 0\) on (0,1) and thus the result follows.
Case (ii). When \(Y(t)>0\). We may write
Thus, to prove that \(r(t)\le 0\), it is enough to prove that \(B(t)\) is an increasing function of \(t\). Now
For \(Y(t)>0\), \(B'(t)\ge 0\) is equivalent to
Now, following three possibilities arise:
-
(a).
If \(Y(t)>0\) throughout the interval (0,1), then (4.4) implies that \(B'(t)\ge 0\) on (0,1). Thus, \(B(t)\) is increasing in (0,1) which implies that, \(B(t)\le B(1)=0\). Therefore, \(r(t)\le 0\) on (0,1).
-
(b).
If \(Y(t)>0\) on some interval \((0,t_{0})\) and \(Y(t)\le 0\) on \([t_{0},1)\) for some \(t_{0}\in (0,1)\), then (4.4) implies that \(B'(t)\ge 0\) on \((0,t_{0})\). Thus, \(B(t)\) is increasing in \((0,t_{0})\) which implies that, \(B(t)\le B(t_{0})\) for any \(t\) in \((0,t_{0})\). Since \(B(t_{0}) \rightarrow -\infty \), this implies that \(B(t)\) is negative. Therefore, \(r(t)\le 0\) on \((0,t_{0})\). In view of Case (i), \(r(t)\le 0\) whenever \(Y(t)\le 0\). Thus, \(r(t)\le 0\) on (0,1).
-
(c).
If \(Y(t)\le 0\) on some interval \((0,t_{0}]\) and \(Y(t)>0\) on \((t_{0},1)\) for some \(t_{0}\in (0,1)\), then (4.4) implies that \(B'(t)\ge 0\) on \((t_{0},1)\). Thus, \(B(t)\) is increasing in \((t_{0},1)\) which implies that, \(B(t)\le B(1)=0\) for any \(t\) in \((t_{0},1)\). Therefore, \(r(t)\le 0\) on \((t_{0},1)\). In view of Case (i), \(r(t)\le 0\) whenever \(Y(t)\le 0\) which implies that, \(r(t)\le 0\) on (0,1).
Subcase (i). For \(\delta =0\), \(X(t)\) and \(Y(t)\) reduces to the simple form
Clearly \(Y(t)\le 0\) on (0,1) if \(\displaystyle \frac{1}{\nu }-2-\frac{1}{\mu }\le 0\) or simply \(\displaystyle \nu \ge {\mu }/{(2\mu +1)}\) and so \(r(t)\le 0\) in this case. On the other hand, if \(\displaystyle 0<\nu <{\mu }/{(2\mu +1)}\) on (0,1), then \(Y(t)>0\) on (0,1) and thus (4.4) gives that
on (0,1) and hence \(r(t)\le 0\) throughout the interval (0,1).
In the case when \(\gamma =0\), we have \(\mu =0\), \(\nu =\alpha >0\). Let
To apply Theorem 2.3 in [9] along with Theorem 3.1, the function \(P(t)=\frac{k(t)}{(1+t)(1-t)^{1+2\delta }}\) must be shown decreasing on the interval (0,1). Since, \(P(t)>0\) on (0,1) and
thus, \(P(t)\) is decreasing in (0,1) provided
Since, \(Q(1)=0\), thus \(Q(t)\le 0\) will certainly hold if \(Q\) is increasing on (0, 1). Now \(\displaystyle Q'(t)=\frac{(1+t)}{2(t+\delta (1+t))^{2}}\left\{ (1-t)(t+\delta (1+t))k''(t)+[2\delta (t+\delta (1+t))\right. \left. -(1-t)(1+\delta )]k'(t)\right\} ,\)
where \(\displaystyle (1-t)(t+\delta (1+t))k''(t)+[2\delta (t+\delta (1+t))-(1-t)(1+\delta )]k'(t)\) is equal to
Thus, \(Q'(t)\ge 0\), for \(t\in (0,1)\), is equivalent to the inequality
The latter condition is equivalent to \(\Delta (t)\ge 0\), where
A simple computation along with (4.3) shows that \(\Delta \) can be rewritten as
Since \({\Lambda }_{\alpha }(t)\ge 0\) and setting
\(\Delta \ge 0\) follows from
Since \(X(t)\) is non-negative on (0,1), thus the inequality \(\Delta \ge 0\) follows from
For \(\delta =0\), (4.6) reduces to
These observations for \(\delta =0\) lead to the following result by, Ali et al. [1], Theorem 4.2].
Corollary 4.1
Assume that both \(\Pi _{\mu ,\nu }(t)\) and \(\Lambda _{\nu }(t)\), as defined in Theorem 3.1 are integrable on [0,1], and positive on (0,1). Let \(\lambda (t)\) be a normalized non-negative real-valued integrable function on [0,1]. Under the same conditions as stated in Theorem 3.1, if \(\lambda \) satisfies
then \(F(z)=V_{\lambda }(f)(z)\in S^{*}\). The conclusion does not hold for smaller values of \(\beta \).
Subcase (ii). If \(0<\delta \le 1/2\) with \(\gamma >0\), then (4.4) can be rewritten as
Since \( Y(t)=X(t)({1/{\nu }}-1-{1/{\mu }})+Z(t)\), so the above inequality is equivalent to
Define \(D(t)=t(1+\delta )-(1-\delta )\). Rewriting the expressions for \(X(t)\) and \(Z(t)\) in terms of \(D(t)\), we get
and so a simple computation gives that
Since \(D^{2}(t)\le 1\) for \(t\in [0,1]\) thus (4.9) is non-negative in (0,1). Since \(X(t)+Z(t)\) and \(X(t)\) are non-negative on (0,1), so if \(\displaystyle \left( {1}/{\nu }-1-{1}/{\mu }\right) \le 0\) or simply \(\displaystyle \nu \ge {\mu }/{(\mu +1)}\), then the inequality (4.8) holds on the interval where \(Y(t)>0\) and hence, \(r(t)\le 0\) on (0,1).
While on the other hand, for \(0<\delta \le 1/2\) with \(\gamma =0\), from (4.6) we have
Since \(X(t)\) and \(X(t)+Z(t)\) are non-negative on (0,1), thus equivalently,
Hence, for \(0<\delta \le 1/2\) with \(\gamma =0\), we have \(\Delta \ge 0\) throughout the interval (0,1).
Thus, we see that above Corollary continues to hold for \(\delta \in (0,1/2]\) but with some restrictions. More precisely, we have
Theorem 4.2
Let \(\lambda (t)\) be a non-negative real-valued integrable function on [0,1]. Assume that both \(\Pi _{\mu ,\nu }(t)\) and \(\Lambda _{\nu }(t)\) are integrable on [0,1], and positive on (0,1). Let \(\lambda \) satisfying the condition
Let \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\) with \(\nu \ge {\mu }/{(\mu +1)}\), and \(\beta <1\) with
where \(g(t)\) is defined by (2.6) with \(\delta \in (0,1/2]\).Then \(F(z)=V_{\lambda }(f)(z)\in S^{*}(\delta )\). The conclusion does not hold for smaller values of \(\beta \).
Remark 4.1
5 Applications
In this section, we present a number of applications of Theorem 4.2 for various well-known integral operators. Let \((a)_{n}\) denote the Pochhammer symbol, defined in terms of the Gamma function, by
Define the Gaussian hypergeometric function by
where \(a\), \(b\) and \(c\) are complex numbers with \(c\ne 0,-1,-2,\ldots \). Note that the series \(_{2}F_{1}\) converges absolutely for \(z\in E\). Now let \(\Phi \) be defined by \(\Phi (1-t)=1+\sum _{n=1}^{\infty }b_{n}(1-t)^n\), \(b_n\ge 0\) for \(n\ge 1\), and
where \(K\) is a constant chosen such that \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). The following result holds in this instance.
Theorem 5.1
Let \(a\), \(b\), \(c\), \(\alpha >0\), \({\nu }\ge {\mu }/(\mu +1)\) and \(\beta <1\) satisfy
where \(K\) is a constant such that \(\displaystyle K\int _{0}^{1}t^{b-1}(1-t)^{c-a-b}{\Phi }(1-t)\mathrm{d}t=1\) and \(g\) is given by (2.6). Then for \(\delta \in [0,1/2]\), we have \(V_{\lambda }({{\mathcal {W}}}_{\beta }(\alpha , \gamma )) \subset S^{*}(\delta )\) provided the following condition hold
where
The value of \(\beta \) is sharp.
Proof
Using (5.1), we have
The condition (4.10) is satisfied when
Since \(\Phi (1-t)=1+\sum _{n=1}^{\infty }b_{n}(1-t)^n\), \(b_n\ge 0\) for \(n\ge 1\), so the functions \(\Phi (1-t)\) and \({\Phi }'(1-t)\) are non-negative in (0,1). Therefore, a simple computation of \((b-1)-\frac{(c-a-b)t}{1-t}\) with \(c-a-b\ge 0\), implies that the condition (4.10) is satisfied whenever \(b\) satisfies (5.2). Hence the result follows by applying Theorem 4.2. \(\square \)
Writing \(\gamma =0\), \(\alpha >0\) in Theorem 5.1 leads to the following corollary:
Corollary 5.1
Let \(a\), \(b\), \(c\), \(\alpha >0,\) and \(\beta <1\) satisfy
where \(K\) is a constant such that \(\displaystyle K\int _{0}^{1}t^{b-1}(1-t)^{c-a-b}{\Phi }(1-t)\mathrm{d}t=1\) and \(g_{\alpha }\) is given by (2.7). If \(f\in {\mathcal {W}}_{\beta }(\alpha , 0)\equiv {\mathcal {P}}_{\alpha }(\beta )\), then the function
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\) whenever \(a\), \(b\), \(c\) are related by \(c\ge a+b\) and \(b\le 4-\frac{1}{\alpha }\), \(\alpha \in (1/4,1/3]\), for all \(t\in (0,1)\). The value of \(\beta \) is sharp.
Writing \(\alpha =1+2\gamma \), \(\gamma >0\) and \(\mu =1\) in Theorem 5.1 gives the following corollary, which is an improvement of the Theorem 4.3 in [3]:
Corollary 5.2
Let \(a\), \(b\), \(c>0\), \(\gamma \ge 1/2\) and \(\beta <1\) satisfy
where \(K\) is constant such that \(\displaystyle K\int _{0}^{1}t^{b-1}(1-t)^{c-a-b}{\Phi }(1-t)\mathrm{d}t=1\) and \(g_{\gamma }\) is given by (2.7). If \(f\in {\mathcal {W}}_{\beta }(1+2\gamma , \gamma )\), then the function
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\) whenever \(a\), \(b\), \(c\) are related by \(c\ge a+b\) and \(0<b\le 3\), for all \(t\in (0,1)\) and \(\gamma >1/2\). The value of \(\beta \) is sharp.
The following special case of Theorem 5.1 corresponds to Bernardi operator, which we state as a theorem.
Theorem 5.2
Let \(c>-1\), \(\nu \ge {\mu }/(\mu +1)\) and \(\beta <1\) satisfy
where \(g\) in given by (2.6). If \(f\in {\mathcal {W}}_{\beta }(\alpha , \gamma )\), then the Bernardi Transform
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\) if
The value of \(\beta \) is sharp.
Taking \(\gamma =0\), \(\alpha >0\) Theorem 5.2 reduces to the following corollary:
Corollary 5.3
Let \(-1<c\le 3-1/{\alpha }\), \(\alpha \in (1/4,1/3]\) and \(\beta <1\) satisfy
where \(g_{\alpha }\) is given by (2.7). If \(f\in {\mathcal {W}}_{\beta }(\alpha , 0)\equiv {\mathcal {P}}_{\alpha }(\beta )\), then the function
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\). The value of \(\beta \) is sharp.
Remark 5.1
For \(\alpha =1+2\gamma \), \(\gamma >0\) and \(\mu =1\) in Theorem 5.2 yields Corollary 4.1 in [3].
To prove the next theorem, we define
where \(b>-1\) and \(a>-1\).
Theorem 5.3
Let \(b>-1\), \(a>-1\), \({\nu }\ge {\mu }/(\mu +1)\) and \(\alpha >0\). Let \(\beta <1\) satisfy
where \(g\) is given by (2.6) and \(\lambda (t)\) is defined by (5.3). If \(f\in {\mathcal {W}}_{\beta }(\alpha , \gamma )\), then the convolution operator
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\) if
The value of \(\beta \) is sharp.
Proof
We omitted the proof as it follows on the same lines as discussed in Theorem 5.3 [1].
Remark 5.2
-
1.
For \(\alpha =1+2\gamma \), \(\gamma >0\) and \(\mu =1\) in Theorem 5.3 yields Theorem 4.1 in [3].
-
2.
For \(\gamma =0\), Theorem 5.3 gives a result similar to Theorem 2.1 [2].
Now, we define
In this case, \(V_{\lambda }\) reduces to the Komatu operator
For \(p = 1\) Komatu operator gives the Bernardi integral operator. For this \(\lambda \), the following result holds.
Theorem 5.4
Let \(-1<a\), \(\alpha >0\), \(p\ge 1\), \({\nu }\ge {\mu }/(\mu +1)\) and \(\beta <1\) satisfy
where \(g\) is given by (2.6). If \(f\in {\mathcal {W}}_{\beta }(\alpha , \gamma )\), then the function
belongs to \(S^{*}(\delta )\) with \(\delta \in (0,1/2]\) if
The value of \(\beta \) is sharp.
Proof
Since
therefore, using the fact that \(\log (1/t)>0\) for \(t\in (0,1)\), and \(p\ge 1\), condition (4.10) is satisfied whenever \(a\) satisfies (5.5).
Remark 5.3
Setting \(\alpha =1+2\gamma \), \(\gamma >0\) and \(\mu =1\) in Theorem 5.4, we get Theorem 4.2 in [3].
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Acknowledgments
The authors are thankful to the learned referees for their useful comments and suggestions which facilitated to improve the present manuscript. The authors understand that the some independent work on similar directions is being carried out by various other researchers e.g., Omar et al. [7].
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Verma, S., Gupta, S. & Singh, S. Duality and Integral Transform of a Class of Analytic Functions. Bull. Malays. Math. Sci. Soc. 39, 649–668 (2016). https://doi.org/10.1007/s40840-015-0131-3
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DOI: https://doi.org/10.1007/s40840-015-0131-3