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On strongly starlike and strongly convex functions with bounded radius and bounded boundary rotation
Journal of Inequalities and Applications volume 2020, Article number: 126 (2020)
Abstract
In this paper, we introduce and investigate new classes of normalized analytic functions in an open unit disk with bounded radius and bounded boundary rotation by using the subordination. We discuss inclusion results, co-efficient bounds, growth and distortion theorems of the classes. Moreover, we compute the radii of strong starlikeness, convexity and starlikeness of the classes. It is interesting to mention that most of our findings are best possible as compared to the existing results in the literature.
1 Introduction and preliminaries
We represent \(\mathcal{A}\) as the class of functions \(f(z)\), which is analytic in \(\mathtt{E}= \{z\in C: \vert z \vert <1 \}\). Then, for each \(z\in E\), \(f(z)\) has the form
Suppose that \(\mathtt{g}(z)\) and \(\mathtt{G}(z)\) are analytic functions in E. Then \(\mathtt{g}(z)\) is said to be subordinate to \(\mathtt{G}(z)\) written as \(\mathtt{g}(z)\prec \mathtt{G}(z)\) if and only if there exists \(w(z)\), which is analytic in E with \(w(\mathtt{0})= \mathtt{0}\) and \(\vert w(z) \vert <1\). Therefore, \(\mathtt{g}(z)= \mathtt{G}(w(z))\) belongs to E. \(\mathtt{g}(z)\) is univalent in E follows that \(\mathtt{g}\prec \mathtt{G}\) is equivalent to \(\mathtt{g}( \mathtt{0})=\mathtt{G}(\mathtt{0})\) and \(\mathtt{g}( E)\subset \mathtt{G}(\mathtt{E})\).
The function \(F(A,B,z)= ( \frac{1+Az}{1+Bz} ) \) is the conformal mapping of the unit disk to circle, which is symmetric with respect to the real axis having center \(\frac{1-AB}{1-B^{2}}\) and radius \(\frac{A-B}{1-B^{2}}\) for each A, B such that \(-1\leq B< A\leq 1\). A function \(p(z)\) (with \(p(\mathtt{0})=1\)) is analytic in E and belongs to the class \(P[A,B]\) if \(p(z)\) is subordinate to \(F(A,B,z)\). In [5], Janowski introduced and investigated the class \(P[A,B]\). Later on, Noor and Arif [8] investigated the class \(P_{m}[A,B]\). They investigated that a function \(p(z)\) (with \(p( \mathtt{0})=1\)) is analytic in E and belongs to the class \(P_{m}[A,B]\) if and only if there exist \(p_{1}(z), p_{2}(z)\in \mathrm{P}[A,B]\) such that
for \({m}\geq 2\), \(-1\leq B< A\leq 1\), and \(z\in E\).
In [14], Rajapat et al. used the known family of fractional integral operators (with the Gauss hypergeometric function in the kernel) and defined new subclasses of strongly starlike and strongly convex functions of order β and type α in the open unit disk U. Moreover, they established several inclusion relationships and interesting results associated with the fractional integral operators.
Shiraishi et al. [16] investigated some new sufficient conditions for the class of strong Caratheodory functions in the open unit disk U. Example and several corollaries of the main results were presented.
We consider a function \(F_{\beta }(A,B,z)= ( \frac{1+Az}{1+Bz} ) ^{\beta }\) for \(-1\leq B\leq 1\), \(-1\leq A\leq 1\) (\(A\neq B\)), and \(0<\beta \leq 1\), which is analytic and univalent in E. If \(p(z)\) (with \(p(\mathtt{0})=1\)) is subordinate to \(F_{\beta }(A,B,z)\), then the function \(p(z)\) is analytic in E and belongs to the class \({\tilde{\mathrm{P}}}^{\beta }[A,B]\).
Lemma 1.1
([10])
Suppose\(f\in \mathcal{V}_{m} ( \rho ) \)for\(m \geq 2\)and\(\mathtt{0} \leq \rho <1 \). Then\(f ( z ) = ( f_{1} ( z ) ) ^{ ( 1-\rho ) }\)for\(f_{1} ( z ) \in \mathcal{V}_{m}\).
Lemma 1.2
([6])
Suppose\({\xi }={\xi }_{1}+ i{\xi }_{2}\), \({\zeta }={\zeta }_{1}+i {\zeta }_{2}\)and\(\varPhi :D\subset C^{2}\rightarrow C\)is a complex-valued function, which satisfies the following conditions:
- (i)
\({\varPsi } ( {\xi },{\zeta } ) \)is continuous in a domain\(D\subset C^{2}\),
- (ii)
\(( 1,\mathtt{0} ) \in \mathsf{D}\)and\({\varPhi } ( 1,\mathtt{0} ) >\mathtt{0}\),
- (iii)
\(\operatorname{Re}{\varPhi } ( i{\xi }_{2},{\zeta }_{1} ) \leq \mathtt{0}\), whenever\(( i{\xi }_{2},{\zeta }_{1} ) \in \mathsf{D}\)and\({\zeta }_{1}\leq -\frac{1}{2} ( 1+{\xi }_{2}^{2} ) \).
If\(q(z)=1+\sum_{n=1}^{\infty }c_{n}z^{n}\)is an analytic function inEwith\(( q(z),zq^{\prime }(z) ) \in D\)and\(\operatorname{Re}\varPhi (q(z), zq^{\prime }(z))\)for\(z\in E\), then\(\operatorname{Re}(q(z))>0\), for\(z\in E\).
Lemma 1.3
([3])
Suppose\(\mathsf{r}_{a}\leq \operatorname{Re}(a)\sin ( \frac{\alpha \pi }{2} ) -\operatorname{Re}(a) \cos ( \frac{\alpha \pi }{2} ) \), \(\operatorname{Im}(a)\geq 0\). Then the disk\(\vert w-a \vert \leq \mathsf{r} _{a}\)is contained in the sector\(\vert \arg (w) \vert \leq \frac{\alpha \pi }{2}\)for\(\mathtt{0}\leq \alpha <1\).
2 Main results
In this section, we discuss the coefficient problem, analytic property, inclusion results, and radius problem. It is interesting to mention that our obtained results are sharp as compared to the existing results in the literature.
First we define the following.
Definition 2.1
A function \(q(z)\) (with \(q(\mathtt{0})=1\)) is analytic in E and belongs to the class \(\tilde{P}_{m}^{ \beta }[A,B]\) (for \({m}\geq 2\)) if and only if there exist \(p_{1}(z), p_{2}(z)\in {\tilde{\mathrm{P}}}^{\beta }[A,B]\) such that
It is easy to see that the set \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\) is convex. For special values of A, B, and β, we obtain several subclasses of analytic functions studied and investigated by several researchers [4, 7–9, 13–15].
Definition 2.2
An analytic function \(f(z )\in \tilde{\mathsf{R}}_{m}^{\beta }[A,B]\) if and only if
Note that, for \(\beta =1\), the class \(\tilde{\mathsf{R}}_{m}^{\beta }[A,B]\) yields the class \(\mathsf{R}_{m}[A,B]\) [8] and \(\tilde{\mathsf{R}}_{m}^{\beta }[1,-1]=\mathsf{R}_{m}(\beta )\). For \(\beta =\frac{1}{2}\), \(B= \mathtt{0}\), and \(A=a\) with \(a\in (\mathtt{0},1]\), we obtain the class
Definition 2.3
A function \(f\in \mathcal{A}\) belongs to \(\tilde{\mathsf{V}}_{m}^{\beta }[A,B]\) if and only if f is locally univalent and
From (2.1) and (2.3), it is considered that
Note that, for particular cases of A, B, and β, several classes are obtained and investigated in [8]. For \(\beta =\frac{1}{2}\), \(m=2\), \(B=\mathtt{0}\) and \(A=a\), \(a\in (\mathtt{0},1]\), Aouf [1] obtained the following class:
Now we prove the following lemmas, which represent the coefficient inequality of functions belonging to the class \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\). By choosing special values of A, B, and β, we may relate our finding with the existing results. We shall assume \(m\geq 2\), \(-1\leq B< A\leq 1\), \(\mathtt{0}<\beta \leq 1\), unless otherwise stated.
Lemma 2.4
Suppose\(q(z)=1+\sum_{n=1}^{\infty }a_{n}z^{n}\)belongs to\({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\). Then
Proof
Suppose \(q(z)\) belongs to \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\) with \(q(z)=1+\sum_{n=1}^{\infty }a_{n}z^{n}\). Then
If \(q_{1}(z)=1+\sum_{n=1}^{\infty }{b}_{n}z^{n}\) and \(q_{2}(z)=1+\sum_{n=1}^{\infty }{c}_{n}z^{n}\), then
By comparing the coefficients of \(z^{n}\) and using the triangle inequality, we get
Since \(q_{i}(z)\prec ( \frac{1+Az}{1+Bz} ) ^{\beta }=1+ \beta (A-B)z+\cdots\) for \(i=1,2\). Using the well-known result of Rogosinski [15] on subordination, we have \(\vert {b}_{n} \vert \leq \beta \vert A-B \vert \) and \(\vert c_{n} \vert \leq \beta \vert A-B \vert \) for all \(n\geq 1\). This implies that
which is as required. This completes the proof. □
The following lemma extends the inequality bound of functions in the class \(\mathrm{P}_{m}[A,B]\).
Lemma 2.5
If\(q(z)\)belongs to\({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\)and\(z=\mathrm{r}e^{i\theta }\), then
where\(k_{1}= ( 1-(A-B)r-ABr^{2} ) ^{\beta }\), \(k_{2}= ( 1+(A-B)r-ABr^{2} ) ^{\beta }\).
Proof
Suppose \(q(z) \in {\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\). Then
Using the triangle inequality, we have
Since \(q_{1}(z), q_{2}(z)\in {\tilde{\mathrm{P}}}^{\beta }[A,B]\), then
where \(w(z)\) is defined in E, which is an analytic function with \(w{(0)}=0\) and \(\vert w(z) \vert < r\). Using (2.8) in (2.7), we get the right-hand side of (2.5).
Moreover, (2.6) can be written as
Since \(q_{1}(z), q_{2}(z)\in {\tilde{\mathrm{P}}}^{\beta }[A,B]\) and
Using (2.10) in (2.9), we have
This implies that
By simple calculations, we get the left-hand side of (2.5). Hence the proof. □
Lemma 2.6
If a function\(q(z)\in \mathcal{A}\)belongs to\({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\), then\(q(z)\in \mathrm{P}_{m}(\gamma )\), where\(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\)for\(z \in E\).
Proof
Let \(q(z)\) belong to \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\). From (2.6) and (2.8), we have
From this it follows that each \(q_{i}(z)\in \mathrm{P}(\gamma )\) for \(i=1,2\), where \(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\) and \(q(z)\in \mathrm{P}_{m}(\gamma )\) for \(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\). This implies that \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\subseteq \mathrm{P}_{m}(\gamma )\). Hence the proof. □
In the following theorems, we discuss and investigate the coefficient problem, analytic property, inclusion results, and radius problem.
Theorem 2.7
If\(f\in \tilde{\mathsf{V}}_{m}^{\beta }[A,B]\), then there exist\({\sigma }_{1},{\sigma }_{2}\in \overset{\ast }{S}(\gamma )\)with\(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\)such that
Proof
Let \(f(z)\) belong to \(\tilde{\mathsf{V}}_{m}^{\beta }[A,B]\). Then \(q(z)=\frac{(zf^{{\prime }}(z))^{{\prime }}}{f^{{\prime }}(z)}\in {\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\). By Lemma 2.6, \({\tilde{\mathrm{P}}} _{m}^{\beta }[A, B]\subseteq P_{m}(\gamma )\). This implies that \(\frac{(zf^{{\prime }}(z))^{{\prime }}}{f^{{\prime }}(z)}\in P_{m}( \gamma )\) with \(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\). It follows that \(f\in V_{m}(\gamma )\). By Lemma 1.1, it has the form
Brannan [2] showed that each \(\mathtt{g}(\mathsf{z})\in \mathsf{V}_{m}\) has the representation of the form
It has been proved in [12] that each \({\sigma }_{i}\in \overset{\ast }{S}(\gamma )\) is
Using (2.14), (2.15), and (2.16), we get our required result (2.13). This completes the proof. □
Theorem 2.8
Let\(f(z)\)belong to\(\tilde{\mathsf{R}}_{m}^{\beta }[A,B]\)with\(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\)for\(z\in E\). Then
Proof
Let \(f(z)\) belong to \(\tilde{\mathsf{R}}_{m}^{\beta }[A,B]\). Then \(q(z)=\frac{zf^{{\prime }}(z)}{f(z)}\in {\tilde{\mathrm{P}}}_{m}^{ \beta }[A,B]\). Suppose \(q(z)=\frac{zf^{{\prime }}(z)}{f(z)}=1+\sum_{n=1}^{\infty }c_{n}z^{n}\), where \(q(z)\) is defined in E and is analytic with \(q( \mathtt{0})=1\). This implies
The series representation of (2.18) is
Comparing the coefficient of \(z^{n}\), we obtain \(( n-1 ) a_{n}=\sum_{{i}=1}^{n-1}c_{ {i}}a_{n-{i}}\), \(c_{\mathtt{0}}=1\) and hence
Since \(q(z)\in {\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\) and using Lemma 2.4, we get \(\vert c_{{i}} \vert \leq \frac{\beta m}{2}(A-B)\) for all n ≥1. From (2.19), we have \(\vert a_{n} \vert \leq \frac{\beta {m}(A-B)}{2(n-1)}\sum_{ {i}=1}^{n-1} \vert a_{{i}} \vert \). In particular,
For \(n=\mathsf{j}\),
where we use the Pochhammer notation \(( \beta ) _{\mathsf{m}}\) introduced in [11]:
Consider
Using (2.20), we obtain the following:
By induction, we get (2.17). Hence the proof. □
Theorem 2.9
Let\(f(z)\)belong to\(\tilde{\mathsf{V}}_{m}^{\beta }[A,B] \)with\(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\), \(z\in E\). Then
Proof
Let \(f(z)\) belong to \(\tilde{\mathsf{V}}_{m}^{\beta }[A,B]\) with \(f(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}\). Then \(zf^{{\prime }}(z)\in \tilde{\mathsf{R}}_{m}^{\beta }[A,B]\). By using Theorem 2.8, we get
□
Corollary 2.10
For\(A=1\), \(B=-1\), \(f\in \tilde{\mathsf{V}}_{m}^{\beta }[1,-1]\equiv \tilde{\mathsf{V}}_{m}(\beta ) \)and using Theorem 2.9, we obtain
Theorem 2.11
If\(f\in \tilde{\mathsf{V}}_{m}^{\beta }[A,B]\), then\(f\in \tilde{\mathsf{R}}_{m}^{\beta }(\rho )\), where
Proof
Suppose that
where \(q(z)\) is analytic in E with \(q(\mathtt{0})=1\). From (2.21), we have
\(f\in \tilde{\mathsf{V}}_{m}^{\beta }[A,B]\) implies that \(\frac{(zf^{{\prime }}(z))^{{\prime }}}{f^{{\prime }}(z)}\in {\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\) and \({\tilde{\mathrm{P}}}_{m}^{\beta }[A,B]\subseteq \mathrm{P}_{m} (\gamma )\), where \(\gamma =(\frac{1-A}{1-B})^{\beta }\). From this it follows that
Using (2.23) and (2.24), we get
Now we define
with \(\eta _{1}=\frac{1}{1-\rho }\), \(\eta _{2}=\frac{\rho }{1-\rho }\). Using (2.22) with the convolution techniques given by Noor [7], we have
Thus, using (2.25) and (2.26), we have
Since the first two conditions of Lemma 1.2 are obviously satisfied, we satisfy condition (iii) as follows:
where
It is observed that (2.28) gives negative value for \({A}\leq \mathtt{0}\) and \(B\leq {0}\). Therefore, using \(A\leq {0}\), we have
and from \(B\leq \mathtt{0}\), we obtain \(\mathtt{0} \leq \rho <1\). Hence all three conditions of Lemma 1.2 are satisfied. From this it follows that \(q_{i}(z)\in \mathrm{P}\) for \(i=1,2\) and \(z\in E\). Therefore \(q(z)\in \mathrm{P}_{m}\) and \(f\in R_{m}(\rho )\), where ρ is given in (2.29). This completes the proof. □
Corollary 2.12
If\(\beta =\frac{1}{2}\), \(B=0\), \(A=a\), \(a\in (0,1]\)and\(f\in \tilde{\mathsf{V}}_{m}^{\frac{1}{2}}[a,0]\). Then, using Theorem 2.11, we have\(f\in\tilde{\mathsf{R}}_{m}^{\frac{1}{2}}(\rho )\), where
Theorem 2.13
If\(f(z)\)belongs to\(\tilde{\mathsf{R}}_{m}^{\beta }[A,B]\), then\(f(z)\in \mathsf{R}_{m}(\gamma ) \), \(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\)for\(z \in E\).
Proof
\(f(z)\) belongs to \(\tilde{\mathsf{R}}_{m}^{\beta }[A,B]\) implies \(q(z)=\frac{zf^{{\prime }}(z)}{f(z)}\) for some \(q\in\tilde{\mathrm{P}}_{m}^{\beta }[A,B]\). Using Lemma 2.6, we get \(\frac{zf^{{\prime }}(z)}{f(z)}\in \mathrm{P}_{m}(\gamma )\), where \(\gamma = ( \frac{1-A}{1-B} ) ^{\beta }\). Thus \(f(z)\in \mathsf{R}_{m}(\gamma )\) for \(z\in E\). This completes the proof. □
Corollary 2.14
If\(\beta =1\), \(m=2\)and\(f\in S^{\ast }[A,B]\). Then, using Theorem 2.13, we have\(f(z)\in S^{\ast }(\gamma )\), where\(\gamma =(\frac{1-A}{1-B})\).
3 Conclusion
We introduced some new classes of analytic functions with bounded radius and bounded boundary rotation by using the subordination. We discussed inclusion results, coefficient bounds, growth and distortion theorems of the classes. The radii of starlikeness and strong starlikeness of the classes have been computed. It is observed that most of the results are best possible.
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Acknowledgements
Authors are thankful to the Editor-in-Chief Prof. Dr. Gradimir V. Milovanovic and all anonymous reviewers for their valuable comments towards the improvement of the paper. The second author is grateful to Air Marshal Javed Ahmed, HI(M) (Retd), Vice-Chancellor, Air University, Islamabad, Pakistan, for providing the excellent research facilities.
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Hussain, S., Ahmad, K. On strongly starlike and strongly convex functions with bounded radius and bounded boundary rotation. J Inequal Appl 2020, 126 (2020). https://doi.org/10.1186/s13660-020-02391-z
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DOI: https://doi.org/10.1186/s13660-020-02391-z