Abstract
The purpose of the present paper is to introduce subclasses of \(p-\)valent functions defined by linear operator. Inclusion relationships for functions in these subclasses are discussed.
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1 Introduction
Denote by \({\mathbb {A}}(p)\) the class of \(p-\)valent analytic functions of the form:
Aouf [1] defined the class \({\textbf{P}}_{k}(p,\gamma )\) of functions g(z) satisfying \(g(0)=p\) and
which generalizes the classes:
-
(i)
\({\textbf{P}}_{k}(1,\gamma )={\textbf{P}}_{k}(\gamma )\) (see [11, 13] and [7] who proved that it is a convex set);
-
(ii)
\({\textbf{P}}_{k}(1,0)={\textbf{P}}_{k}\) (see [15]);
-
(iii)
\({\textbf{P}}_{2}(p,\gamma )={\textbf{P}}(p,\gamma )\) is the class in which \(\text {Re} \{g(z)\}>\gamma \) \((0\le \gamma <p);\)
-
(iv)
\({\textbf{P}}_{2}(1,\gamma )={\textbf{P}}(\gamma )\) is the class in which \( \text {Re} \{g(z)\}>\gamma \) \((0\le \gamma <1);\)
-
(v)
\({\textbf{P}}_{2}(1,0)={\textbf{P}}\) is the class in which \(\text {Re} \{g(z)\}>\gamma \) .
From (1.2), we have \(g\in {\textbf{P}}_{k}(p,\gamma )\) if and only if there exist \(g_{1},g_{2}\in {\textbf{P}}(p,\gamma )\) such that
Shams et al. [19], defined the integral operator \(I_{p}^{\alpha }:{\mathbb {A}} (p)\rightarrow {\mathbb {A}}(p)\) by:
Note that \(I_{1}^{\alpha }=I^{\alpha }\) was defined by Jung et al. [5].
Taking
we define the function \(G_{p}^{\alpha *}(z)\) by
Using the function \(G_{p}^{\alpha *}(z)\), we define the new linear operator \(N_{p}^{\alpha ,\delta }:{\mathbb {A}}(p)\rightarrow {\mathbb {A}}(p)\) by
where \(\left( v\right) _{n}\) given by
We note that:
It is readily verified from (1.7) that, this linear operator has the two
recurrence relation:
By using each of the recurrence relations (1.8) and (1.9), we obtain different results.
For \(0\le \gamma ,\) \(\beta <p,\) \(p\in {\mathbb {N}} \) and \(k\ge 2\). Seoudy [18]. (see also Noor [8 with \(p=1\)]), defined the following classes by:
We can easily see that:
and
We note that, for special choices for the parameters k and \(\gamma \) involved in the above classes, we can obtain the well-known subclasses:
-
(i)
\({\mathcal {R}}_{2}(p,\gamma )=S_{p}^{*}(\gamma )\) (see [2, 12, 14] and [4]),
- (ii)
-
(iii)
\(T_{2}\left( p,\gamma ,\beta \right) =T_{p}\left( \gamma ,\beta \right) \) (see [3]),
-
(iv)
\(T_{2}^{*}\left( p,\gamma ,\beta \right) =T_{p}^{*}\left( \gamma ,\beta \right) \) (see [9, 10] and [20]).
Next, by using the linear operator \(N_{p}^{\alpha ,\delta }f(z),\) we introduce the following classes of analytic functions for \(0\le \gamma ,\beta <p\) and \(k\ge 2:\)
Note that
and
In particular, we set \({\mathcal {R}}_{1}^{\alpha ,\delta }(k,\gamma )=\mathcal { R}^{\alpha ,\delta }(k,\gamma )\), \(V_{1}^{\alpha ,\delta }(k,\gamma )=V^{\alpha ,\delta }(k,\gamma ),T_{1}^{\alpha ,\delta }\left( k,\gamma ,\beta \right) =T^{\alpha ,\delta }\left( k,\gamma ,\beta \right) \) and \( T_{1}^{*(\alpha ,\delta )}\left( k,\gamma ,\beta \right) =T^{*(\alpha ,\delta )}\left( k,\gamma ,\beta \right) .\)
The following lemma will be required in our investigation.
Lemma 1
[6]. Let \(\phi \left( u,v\right) \) be acomplex valued function,
and let \(u=u_{1}+i\ u_{2}\,v=v_{1}+iv_{2}.\) Suppose that \(\phi \left( u,v\right) \) satisfies the following conditions:
-
(i)
\(\phi \left( u,v\right) \) is continuous in D ;
-
(ii)
\(\ \left( 1,0\right) \in D\) and \(\text {Re} \left\{ \phi \left( 1,0\right) \right\} >0;\)
-
(iii)
for all \(\left( iu_{2},v_{1}\right) \in \) D such that \(v_{1}\le \frac{-\left( 1+u_{2}^{2}\right) }{2},\) \(\text {Re} \left\{ \phi \left( iu_{2},v_{1}\right) \right\} \le 0.\)
Let \(h\left( z\right) =1+h_{1}z+h_{2}z^{2}+....\)be regular in \({\mathbb {U}}\). Such that \(\left( h\left( z\right) ,zh^{^{\prime }}\left( z\right) \right) \in D\) for all \(z\in {\mathbb {U}}.\) If
then \(\text {Re} \left\{ h\left( z\right) \right\} >0\), \(z\in {\mathbb {U}}\) .
Lemma 2
[16]. Let \(p\left( z\right) \) be analytic in \({\mathbb {U}}\) with \(p(0)=a\) and \(\Re \{p(z)\}>0,\) \(z\in {\mathbb {U}}.\) Then for \(s>0\) and \( \mu \in {\mathbb {C}} \backslash \{-1\},\)
where \(r_{0}\) is given by
and this radius is the best possible.
Lemma 3
[17]. Let \(\phi \) be convex and let g be starlike in \( {\mathbb {U}}.\) Then, for F analytic in \({\mathbb {U}}\) with \(F(0)=1,\left( \left( \phi *Fg\right) /\left( \phi *g\right) \right) \) is contained in the convex hull of \(F(\mathbb {U)}.\)
In this paper, we obtain several inclusion properties of the classes \( {\mathcal {R}}_{p}^{\alpha ,\delta }(k,\gamma ),\) \(V_{p}^{\alpha ,\delta }(k,\gamma ),\) \(T_{p}^{\alpha ,\delta }\left( k,\gamma ,\beta \right) \) and \( T_{p}^{*(\alpha ,\delta )}\left( k,\gamma ,\beta \right) \) associated with the operator \(N_{p}^{\alpha ,\delta }f(z).\)
2 Main results
Unless otherwise mentioned, we assume throughout this paper that \(k\ge 2,\) \( 0\le \gamma <p,\) \(p\in {\mathbb {N}} \) , \(\delta >-p\) and \(\alpha \mathbb {>}0.\)
Theorem 1
Let \(0\le \gamma _{1}\le \gamma <p\), then
where
Proof
Let \(f(z)\in {\mathcal {R}}_{p}^{\alpha +1,\delta }(k,\gamma )\) and
where \(h_{i}\) is analytic in \({\mathbb {U}}\) with \(h_{i}(0)=1,i=1,2.\) Using the identity (1.8) in (2.3) and differentiating the resulting equation we obtain
Now, we will show that \(H(z)\in {\textbf{P}}_{k}(p,\gamma _{1})\) or \( h_{i}(z)\in P.\) From (2.3) and (2.4) we have
this implies that
We form the functional \(\phi (u,v)\) by taking \(u=h_{i}(z),\) \( v=zh_{i}^{^{\prime }}(z),\)
Clearly, the first two conditions of Lemma 1 are satisfied in the domain \( D\subseteq {\mathbb {C}} \backslash \frac{\left( \gamma _{1}+1\right) }{\gamma _{1}-p}\times {\mathbb {C}} \). Now, we verify condition (iii) as follows:
where
We note that \(\text {Re} \left\{ \phi \left( iu_{2},v_{1}\right) \right\} <0\) if and only if \(A\le 0\) and \(B<0.\) From \(\gamma _{1}\) as given by (2.2), we obtain \(A\le 0\) and from \(0\le \gamma _{1}\le \gamma <p\) we have \(B<0.\) Therefore applying Lemma 1, \(h_{i}\in {\textbf{P}}(i=1,2)\) and consequently \( H(z)\in {\textbf{P}}_{k}(p,\gamma _{1})\) for \(z\in {\mathbb {U}}.\) \(\square \)
Theorem 2
Let \(0\le \gamma _{1}\le \gamma <p\), then
where
Proof
The proof of Theorem 2 is the same as the proof of Theorem 1 by using (1.9) instead of (1.8). \(\square \)
Theorem 3
Let \(0\le \gamma _{1}\le \gamma <p\) , \( \delta >-p\) and \(k\ge 2\) then
where \(\gamma _{1}\) is given by (2.2).
Proof
Applying (1.20) and Theorem 1, we observe that
which evidently prove Theorem 3. \(\square \)
Similarly, we can prove the following result.
Theorem 4
Let \(0\le \gamma _{1}\le \gamma <p\) , \( \delta >-p\) and \(k\ge 2\) then
where \(\gamma _{1}\) is given by (2.8).
Theorem 5
Let \(\alpha \ge 0\), \(\delta >-p,\) \(0\le \gamma ,\beta <p\) and \(k\ge 2\) then
Proof
Let \(f(z)\in T_{p}^{\alpha +1,\delta }\left( k,\gamma ,\beta \right) \). Then, in view of the definition of the class \(T_{p}^{\alpha +1,\delta }\left( k,\gamma ,\beta \right) ,\) there exists a function \(g(z)\in \mathcal { R}_{p}^{\alpha +1,\delta }(2,\gamma )\) such that
Now let
where h(z) is analytic in \({\mathbb {U}}\) with \(h(0)=1.\) Using (1.8) in (2.12), we have
Differentiating (2.13) leads to
Since \(g(z)\in {\mathcal {R}}_{p}^{\alpha +1,\delta }(2,\gamma ),\) by Theorem 1, \(g(z)\in {\mathcal {R}}_{p}^{\alpha ,\delta }(2,\gamma )\), then we have
where \(p(z)=1+c_{1}z+c_{2}z^{2}+...\) is analytic in \({\mathbb {U}}\) with \( p(0)=1.\) Then by using (1.8), we have
From (2.14) and (2.15), we obtain
Now, we will show that \(H(z)\in {\textbf{P}}_{k}(p,\beta )\) or \(h_{i}\in {\textbf{P}}(i=1,2).\) From (2.3) and (2.16) we have
this implies that
We form the functional \(\phi (u,v)\) by choosing \(u=h_{i}(z),\) \( v=zh_{i}^{^{\prime }}(z),\)
Clearly, conditions (i) and (ii) of Lemma 1 are satisfied in \(D\subseteq {\mathbb {C}} \backslash Q^{*}\times {\mathbb {C}} \), where \(Q^{*}=\left\{ z\in {\mathbb {C}} \text { and }\text {Re} (p(z))=p_{1}>\frac{\gamma +1}{\gamma -p}\right\} \) and \( \ p(z)=p_{1}+ip_{2}.\)
Now, we verify condition (iii) as follows:
By applying Lemma 1, \(h_{i}(z)\in {\textbf{P}}(i=1,2)\). \(\square \)
Theorem 6
Let \(\alpha \ge 0\) , \(\beta <p,\) \(\delta >-p,0\le \gamma <p\) and \(k\ge 2\) then
Proof
The proof of Theorem 6 is the same as the proof of Theorem 5 by using (1.9) instead of (1.8). \(\square \)
Theorem 7
Let \(\alpha \ge 0\) , \(\beta <p,\) \(\delta >-p,0\le \gamma <p\) and \(k\ge 2\) then
Proof
By applying (1.21) and Theorem 5, it follows that
\(\square \)
Similarly, we can prove the following result.
Theorem 8
Let \(\alpha \ge 0\) , \(\beta <p,\) \(\delta >-p,0\le \gamma <p\) and \(k\ge 2\) then
Theorem 9
If \(f(z)\in {\mathcal {R}}_{p}^{\alpha ,\delta +1}(k,0),\) for \(z\in {\mathbb {U}},\) then \(f\in {\mathcal {R}}_{p}^{\alpha ,\delta }(k,0)\) for
where \(A=2(s+1)^{2}+\left| \mu \right| ^{2}-1,\) \(\mu \ne -1\) with \( \mu =1\) and \(s=1.\) This radius is the best possible.
Proof
We begin by setting
where \(h_{i}\) is analytic in \({\mathbb {U}}\) with \(h_{i}(0)=1,i=1,2.\) Using a similar argument as in Theorem 1, we obtain
Applying Lemma 2, we get
where \(r_{0}\) is given by (2.20). \(\square \)
Theorem 10
Let \(\Psi \) be a convex function and \(f(z)\in \mathcal {R }_{p}^{\alpha ,\delta }(2,\gamma ).\) Then \(G\in {\mathcal {R}}_{p}^{\alpha ,\delta }(2,\gamma ),\) where \(G=\Psi *f.\)
Proof
Let \(G=\Psi *f\) , where f(z) is given by (1.1) and
Then
Also, \(f(z)\in {\mathcal {R}}_{p}^{\alpha ,\delta }(2,\gamma ).\) Therefore \( N_{p}^{\alpha ,\delta }f(z)\in {\mathcal {R}}_{2}(p,\gamma ).\) By logarithmic differentiation of (2, 25), we obtain
where \(F=z\left( N_{p}^{\alpha ,\delta }G(z)\right) ^{^{\prime }}/p\left( N_{p}^{\alpha ,\delta }G(z)\right) \) is analytic in \({\mathbb {U}}\) and \( F(0)=1. \) From Lemma 3, we can see that \(z\left( N_{p}^{\alpha ,\delta }G(z)\right) ^{^{\prime }}/p\left( N_{p}^{\alpha ,\delta }G(z)\right) \) is contained in the convex hull of \(F({\mathbb {U}}).\) Since \(z\left( N_{p}^{\alpha ,\delta }G(z)\right) ^{^{\prime }}/p\left( N_{p}^{\alpha ,\delta }G(z)\right) \) is analytic in \({\mathbb {U}}\) and
then \(z\left( N_{p}^{\alpha ,\delta }G(z)\right) ^{^{\prime }}/p\left( N_{p}^{\alpha ,\delta }G(z)\right) \) lies in \(\Omega ,\) this implies that \( G=\Psi *f\in {\mathcal {R}}_{p}^{\alpha ,\delta }(2,\gamma ).\)
\(\square \)
Remark
Specealizing the parameters \(\delta ,\alpha \) and p in the above results, we obtain results concerning the operators \(N_{p}^{\alpha }f(z)\) and \(N^{\alpha ,\delta }f(z)\) given in the introduction.
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The authors are thankful to the referees for helpful suggestions.
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Aouf, M.K., Mostafa, A.O. & El-Hawsh, G.M. Inclusion properties for classes of \(p-\)valent functions associated with linear operator. Afr. Mat. 35, 19 (2024). https://doi.org/10.1007/s13370-023-01149-2
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DOI: https://doi.org/10.1007/s13370-023-01149-2