Abstract
We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions.
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1 Introduction
A continuous function \(f=u+iv\) is a complex-valued harmonic function in a domain \(D {\subset } \mathbb {C}\) if both u and v are real harmonic in D. In any simply connected domain D, we write \(f=h+ \overline{g}\), where h and g are analytic in D. A necessary and sufficient condition for f to be locally one-to-one and orientation preserving in D is that \(|g^{\prime }(z)|<|h^{\prime }(z)|\) for \({z}\in D\) (see Clunie and Sheil-Small [4]). Functions that are harmonic and univalent in \({\mathbb {D}}=\{z:\left| {z}\right| >{1}\}\) are investigated by Hengartner and Schober [7]. In particular, it was shown in [7] that a complex-valued, harmonic, orientation preserving univalent mapping f, defined on \(\mathbb {D}\) and satisfying \(f(\infty )=\infty \), must admit the representation
where \(h(z)={\alpha }z+\sum _{n=0}^{\infty }a_{n}z^{-n}\), \(g(z)={\beta } z+\sum _{n=0}^{\infty }b_{n}z^{-n}\), \(0\ {\le }\ |\beta |<|\alpha |\), and \({ \omega }={\overline{f}}_{\overline{z}}/f_{z}\) is analytic and satisfies \(|{ \omega }(z)|<1\) for \({z}\ \in \ {\mathbb {D}}\). We remove the logarithmic singularity in (1) by letting \(A=0\) and also let \({\alpha }=1\) and \({ \beta }=0\) and focus on the family \({\Sigma }_{\mathcal {H}}\) of meromorphic harmonic orientation preserving univalent mappings of the form
where
We say that a function \(f\in \Sigma _{\mathcal {H}}\) is harmonic starlike in \( \mathbb {D}\) if
or
where \(z = re^{it} {\in } {\mathbb {D}}\), \(0 {\le } t {\le } 2\pi \), and
For \(l=1,2\) and functions \(f_{l}\in \Sigma _{\mathcal {H}}\) of the form
we definethe convolution of \(f_{1}\) and \(f_{2}\) by
For \(m\in \mathbb {N}_{0}:=\left\{ 0\right\} {\cup } {\mathbb {N}}=\left\{ 1,2,...\right\} \) and \(f=h+{\overline{g}} {\in } {\Sigma }_{\mathcal {H}}\) we define the linear operator \(D_{\mathcal {H}}^{m}:{\Sigma }_{\mathcal {H} }\rightarrow {\Sigma }_{\mathcal {H}}\) by
where
We note that \(D_{\mathcal {H}}^{0}f=f\) and \(D_{\mathcal {H}}^{1}f=D_{\mathcal {H}}f\).
In 2008 Mauir [13] considered a weak subordination for complex-valued harmonic functions defined in the open unit disk \(\Delta :=\{z:\left| {z} \right| <{1}\}\). In [8] Jahangiri (see also [10]) investigated the classes of harmonic meromorphic starlike and convex functions of order \( \gamma .\) To obtain some generalizations of these classes we introduce definition of weak subordination for complex-valued functions in \(\mathbb {D} .\)
A complex-valued function f in\(\ \mathbb {D}\) is said to be weakly subordinate to a complex-valued function F in\(\ \mathbb {D}\), and we write \(f(z)\preceq F(z)\) (or simply \(f\preceq F\)), if \(f(\infty )=F(\infty )\) and \( f(\mathbb {D})\subset F(\mathbb {D}).\)
If \(f\preceq F\) and F is univalent in \(\mathbb {D}\), then we can consider the function \(\omega \left( z\right) =F^{-1}\left( f\left( z\right) \right) , z\in \mathbb {D}\) which maps \(\mathbb {D}\) into oneself with \( \omega (\infty )=\infty .\) Conversely, if \(\omega \left( z\right) =F^{-1}\left( f\left( z\right) \right) , z\in \mathbb {D},\) maps \(\mathbb {D}\) into oneself with \(\omega (\infty )=\infty \), then \(f\preceq F.\) Thus, we have the following equivalence.
Lemma 1
A complex-valued function f in \( \mathbb {D}\) is weakly subordinate to a function complex-valued function F in \( \mathbb {D}\) if and only if there exists a complex-valued function \(\omega \) which maps \(\mathbb {D}\) into oneself with \(\omega (\infty )=\infty \) such that \(f(z)=F(\omega (z)), z\in \mathbb {D}.\)
The following two subclasses of \({\Sigma }_{\mathcal {H}}\) are the main focus of this paper.
For \(-B\le A<B\le 1\) let \(\Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) be the class of functions \(f\in \Sigma _{\mathcal {H}}\) so that
and \({\mathbb {V}}_{\mathcal {H}}^{m}\left( A,B\right) \) be the class of functions \(f {\in } \Sigma _{\mathcal {H}}\) so that
In particular, for \(m=0,m=1,\) we obtain classes studied in [5]. The classes \(\Sigma _{\mathcal {H}}^{*}(\gamma ):=\Sigma _{\mathcal {H} }^{1}(2\gamma -1,1) \)and\( \Sigma _{\mathcal {H}}^{c}(\gamma ):=\Sigma _{ \mathcal {H}}^{1}(2\gamma -1,1)\) are investigated by Jahangiri [8]. The classes \(\Sigma _{\mathcal {H}}^{*}:=\Sigma _{\mathcal {H}}^{*}(0)\) and \(\Sigma _{\mathcal {H}}^{c}:=\Sigma _{\mathcal {H}}^{c}(0)\) are the classes of functions \(f\in \Sigma _{\mathcal {H}}\left( k\right) \) which are starlike in \({\mathbb {U}}\left( r\right) \) or convex in \({\mathbb {U}}\left( r\right) ,\) respectively, for all \(r>1\) (see [10]).
We further let \(\mathcal {T}_{\eta }\) be the class of functions \(f=h+ \overline{g}\in {\Sigma }_{\mathcal {H}}\) with varying coefficients (e.g. see [9]) for which there exists a real number \(\eta \) so that
and \(\Sigma _{\eta }^{m}(A,B):=\mathcal {T}_{\eta }\cap \Sigma _{\mathcal {H} }^{m}\left( A,B\right) \) and \(V_{\eta }^{m}(A,B):=\mathcal {T}_{\eta }\cap V_{ \mathcal {H}}^{m}\left( A,B\right) .\)
2 Dual sets and coefficient bounds
Here we make use of the “duality principle” defined by Ruscheweyh ([14] , Chapter 1). For the set \(\mathcal {V}\ {\subset } {\Sigma }_{\mathcal {H}}\) we define the dual set of \(\mathcal {V}\) by
Consequently, the second dual of \(\mathcal {V}\) or the dual of \({\mathcal {V}} ^{*}\) is defined as
This duality principle indicates that under fairly weak conditions on \( \mathcal {V}\) which is a subset of \({\Sigma }_{\mathcal {H}}\), many linear and other extremal problems in the second dual of \(\mathcal {V}\) are solved in \( \mathcal {V}\). This is a very useful tool since in many cases of interest (such as convex, starlike or close-to-convex harmonic univalent functions), the set \({\mathcal {V}}^{{*}{*}}\) is much larger than the set \(\mathcal { V}\). This, on its own right, would be a separate endeavor and research topic that can be explored further which is not the focus of the present paper. The following theorem presents a duality condition for the set \({\Sigma }_{ \mathcal {H}}^m(A,B)\).
Theorem 1
where
Proof
Let \(f\in \Sigma _{\mathcal {H}}\). Then \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if
Since
the above inequality (5) yields
Thus, \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if \( f\left( z\right) *\psi _{\xi }\left( z\right) \ne 0\) i.e. \( \Sigma _{\mathcal {H}}^{m}\left( A,B\right) =\left\{ \psi _{\xi }:\ \left| \xi \right| =1\right\} ^{*}\). \(\square \)
A similar argument can be used to obtain a duality condition for the set \({ \mathbb {V}}_{\mathcal {H}}^m(A,B)\).
Theorem 2
where
Next we determine sufficient coefficient bounds for function in \(\Sigma _{\mathcal {H}}^{m}\left( A,B\right) \).
Theorem 3
Let \(f=h+\overline{g}\) be of the form (2) and \(-B\ {\le } \ A<B\ {\le }\ 1\). If
then f is univalent and orientation preserving in \(\mathbb {D}\) and \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \).
Proof
The univalency and orientation preserving of the function f follows by a result of Jahangiri and Silverman ([10], Theorem 1) since
Therefore, by Lemma 1 \(f\in \Sigma _{\mathcal {H}}^{m}\left( A,B\right) \) if and only if there exists a complex-valued function \(\omega \) where \( \omega (\infty )=\infty \), \(\left| \omega (z)\right| >1\) and \(z\in \mathbb {D}\) such that
or equivalently
Thus for \(z\in \mathbb {D}\) it suffices to show that
Indeed, letting \(\left| z\right| =r\,~(r>1)\) we have
where \({\alpha }_n=m_n(n(1+B)+(1+A))\) and \({\beta }_n=m_n(n(1+B)-(1+A)))\). Thus, according to the hypothesis of the theorem, \(f\in \Sigma _{\mathcal {H} }^{m}\left( A,B\right) \). \(\square \)
The sufficient coefficient bound given in Theorem 3 is also necessary for functions to be in the class \(\Sigma _{\eta }^{m}(A,B)\) as stated in the following theorem.
Theorem 4
Let \(f=h+\overline{g}\) be of the form (4) and \(-B\ {\le }\ A<B\ {\le }\ 1\). Then \(f\in \Sigma _{\eta }^{m}(A,B)\) if and only if
Proof
The “if” part follows from Theorem 3. For the “only-if” part, assume that \(f\in \Sigma _{\eta }^{m}(A,B),\) then by (6) we have
Therefore, by (4) for \(z=re^{i\eta }~(r>1),\) we obtain
It is clear that the denominator of the ratio in the inequality (7) cannot vanish for \(r>1.\) Moreover, it is positive as \(r{\rightarrow }\infty \) and consequently for \(r>1.\) Thus, we must have
where \({\alpha }_n=m_n(n(1+B)+(1+A))\) and \({\beta }_n=m_n(n(1+B)-(1+A)))\). The sequence of partial sums \(\left\{ S_{n}\right\} \) associated with the series \(\sum \nolimits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \) is a non-decreasing sequence. Moreover, by (8), it is bounded above by \(B-A.\) Hence, the sequence\(\ \left\{ S_{n}\right\} \) is convergent and
\(\square \)
A similar argument can be used to prove the following theorem.
Theorem 5
Let \(f=h+\overline{g}\) be of the form (2) and \(-B\ {\le } \ A<B\ {\le }\ 1\). Then \(f\in {\mathbb {V}}_{\eta }^{m}(A,B)\) if and only if
As a consequence of Theorems 4 and 5 we have the following corollary.
Corollary 1
For \(-B\ {\le }\ A<B\ {\le }\ 1\) let \(\ a=\frac{1+A}{1+B}\),
Then
3 Extreme points
The Krein-Milman theorem (see [11]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.
Lemma 2
Let \(\mathcal {F}\) be a non-empty compact convex subset of the set \({\Sigma }_{\mathcal {H}}\) and \(\mathcal {J}:{\Sigma }_{\mathcal {H} }\rightarrow \mathbb {R}\) be a real-valued, continuous and convex functional on \(\mathcal {F}.\) Then
where \(E\mathcal {F}\) denotes the set of extreme points of \(\mathcal {F}.\)
Since \({\Sigma }_{\mathcal {H}}\) is a complete metric space, Montel’s theorem [12] implies the following lemma.
Lemma 3
A set \(\mathcal {F}\subset {\Sigma }_{\mathcal {H}} \) is compact if and only if \(\mathcal {F}\) is closed and locally uniformly bounded.
Now we are equipped to state and prove the two main theorems in this section.
Theorem 6
The set \(\Sigma _{\eta }^{m}(A,B)\) is a convex and compact subset of \({\Sigma }_{\mathcal {H}} \).
Proof
For \({l}=1,2\) let \(f_{l}\in \Sigma _{\eta }^{m}(A,B)\) be functions of the form (3),\({\;0\le \gamma \le 1.}\) Since
and since for
(by Theorem 4), we have
the function \({\phi }=\gamma f_{1}+(1-\gamma )f_{2}\) belongs to the class \( \Sigma _{\eta }^{m}(A,B)\). Hence, the set \(\Sigma _{\eta }^{m}(A,B)\) is convex. Furthermore, for \(f\in \Sigma _{\eta }^{m}(A,B)\) and \(\left| z\right| = r>1\) we have
Thus, the class \(\Sigma _{\eta }^{m}(A,B)\) is locally uniformly bounded. Now, by Lemma 3, we only need to show that it is closed i.e. if \(f_{l}\in \Sigma _{\eta }^{m}(A,B) \) and \(f_{l}\rightarrow f\) then \(f\in \Sigma _{\eta }^{m}(A,B).\) Let \(f_{l}\) and f be given by (3) and (4), respectively. Using Theorem 4 we have
Since \(f_{l}\rightarrow f,\) we conclude that \(a_{l,n}\rightarrow a_{n}\) and \( b_{l,n}\rightarrow b_{n}\) as \(l\rightarrow \infty \left( n\in \mathbf { \mathbb {N}}\right) \). The sequence of partial sums \(\left\{ S_{n}\right\} \) associated with the series \(\sum \nolimits _{n=1}^{\infty }\left( \alpha _{n}\left| a_{n}\right| +\beta _{n}\left| b_{n}\right| \right) \) is a non-decreasing sequence. Moreover, by (10), it is bounded above by \(B-A.\) Therefore, the sequence\(\left\{ S_{n}\right\} \) is convergent and
This implies that \(f\in \Sigma _{\eta }^{m}(A,B)\) which completes the proof. \(\square \)
In the following theorem we determine the extreme points of \(\Sigma _{\eta }^{m}(A,B)\).
Theorem 7
where
Proof
Suppose that \(0<\gamma <1\) and \(g_{n}=\gamma f_{1}+\left( 1-\gamma \right) f_{2}\) where \(f_{1},f_{2}\in \Sigma _{\eta }^{m}(A,B)\) are functions of the form (3) and let \(\alpha _{n},\beta _{n}\) be defined by (9). Then \(\left| b_{1,n}\right| =\left| b_{2,n}\right| =\) \(\frac{ B-A}{\beta _{n}}\) and consequently \(a_{1,n}=a_{2,n}=0\) and \( b_{1,k}=b_{2,k}=0 \) for \(k\in \mathbf {\mathbb {N}} \diagdown \left\{ n\right\} .\) It follows that \(g_{n}=f_{1}=f_{2},\) and so \(g_{n}\in \Sigma _{\eta }^{m}(A,B)\). Similarly, we can verify that the functions \(h_{n}\) of the form (11) are the extreme points of the class \(\Sigma _{\eta }^{m}(A,B).\) Now, suppose that the function f belongs to the set \( E\Sigma _{\eta }^{m}(A,B)\) and f is not of the form (11). Then there exists \(s\in \mathbf {\mathbb {N}}\) such that
If \(0<\left| a_{s}\right| <\frac{B-A}{{\alpha }_{s}}\), then setting
we have that \(h_{s}\ne \varphi \) and so \(f=\gamma h_{s}+\left( 1-\gamma \right) \varphi ,\ 0<\gamma <1\). Thus, \(f\notin E\Sigma _{\eta }^{m}(A,B).\)
Similarly, if \(0<\left| b_{s}\right| <\frac{B-A}{\beta _{n}}\), then setting
we have that \(g_{s}\ne \phi \) and so \(f=\gamma g_{s}+\left( 1-\gamma \right) \phi ,\ 0<\gamma <1\).
It follows that \(f\notin E\Sigma _{\eta }^{m}(A,B)\) and so the proof is completed. \(\square \)
4 Applications
It is clear that if the class
is locally uniformly bounded, then
Thus, by Theorem 7, we have the following corollary.
Corollary 2
where \(h_{n}\) and \(g_{n}\) are given by (11).
For \(f\ {\in }\ {\Sigma }_{\mathcal {H}}\), \(\gamma \ge 1\) and \(r>1\) the real-valued functional
is continuous and convex on \({\Sigma }_{\mathcal {H}}\).
Moreover, for \(f\ {\in }\ {\Sigma }_{\mathcal {H}}\), \(z\ {\in }\ {\mathbb {D}}\) and each fixed \(n\ {\in }\ {\mathbb {N}}\) the real-valued functionals \({ \mathcal {J}}_1\left( f\right) =a_{n}\), \({\mathcal {J}}_2\left( f\right) =b_{n} \), \({\mathcal {J}}_3\left( f\right) =\left| f\left( z\right) \right| \) and \({\mathcal {J}}_4\left( f\right) =\left| D_{\mathcal {H} }f\left( z\right) \right| \) are also continuous and convex on \({\Sigma }_{ \mathcal {H}}\).
These in conjunction with Lemma 2 yield the following corollaries.
Corollary 3
Let \(f\in \Sigma _{\eta }^{m}(A,B)\). Then
The result is sharp and the functions \(h_{n}\) and \(g_{n}\) of the form (11) are the extremal functions.
Corollary 4
Let \(f\in \Sigma _{\eta }^{m}(A,B)\). Then
and
The result is sharp and the extremal function is given by (11).
The following covering result follows from Corollary 4.
Corollary 5
If \(f\in \Sigma _{\eta }^{m}(A,B),\) then \(\mathbb {D}\left( 2\right) \subset f\left( \mathbb {D}\right) .\)
Corollary 6
Let \(r>1\) and \(\gamma \ge 1\). If \(f\in \Sigma _{\eta }^{m}(A,B),\) then
Corollary 7
The class \({\mathbb {V}}_{\eta }^{m}(A,B)\) is a convex compact subset of \({ \Sigma }_{\mathcal {H}} \). Moreover,
and
where \(h_{0}(z)=z,\) and
Corollary 8
Let \(f\in {\mathbb {V}}_{\eta }^{m}(A,B)\) be a function of the form (4) . Then
The results are sharp and the functions \(h_{n}\) and \(g_{n}\) of the form (12) are the extremal functions.
Corollary 9
If \(f\in {\mathbb {V}}_{\eta }^{m}(A,B),\) then \(\mathbb {D}\left( r\right) \subset f\left( \mathbb {D}\right) \) where
Remark 1
The classes \({\Sigma }_{\mathcal {H}}^{m}(A,B)\) and \({\mathbb {V}}_{\mathcal {H} }^{m}(A,B)\) for different values of m, A and B give various well-known as well as new classes of meromorphic harmonic univalent functions (see for example [2, 3, 5, 6, 8, 10]).
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The work is supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.
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Dziok, J. Classes of meromorphic harmonic functions and duality principle. Anal.Math.Phys. 10, 55 (2020). https://doi.org/10.1007/s13324-020-00401-3
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DOI: https://doi.org/10.1007/s13324-020-00401-3