1 Introduction
Harmonic functions are famous for their use in the study of minimal surfaces and also play important roles in a variety of problems in applied mathematics (e.g. see Choquet [1], Dorff [2], Duren [3] or Lewy [4]). A continuous function \(f=u+iv\) is said to be complex-valued harmonic in a domain \(D\subset\mathbb{C}\) if both u and v are real harmonic in D. In any simply connected domain, we can write \(f=h+\overline{g}\), where h and g are analytic in D. We shall call h the analytic and g the co-analytic part of f. Clunie and Sheil-Small [5] pointed out that a necessary and sufficient condition for f to be locally univalent and sense preserving in D is that \(\vert h^{\prime} ( z ) \vert >\vert g^{\prime} ( z ) \vert \) in D. Note that for \(f=h+\overline{g}\), harmonic and sense preserving in the open unit disk \(\mathbb{D}=\{z\in \mathbb{C}:\vert {z}\vert <{1\}}\), the condition \(h^{\prime}(0)=1>|g^{\prime}(0)|\) implies that the function \(( f-\overline{g^{\prime}(0)f} ) / ( 1-\vert g^{\prime } ( 0 ) \vert ^{2} ) \) is also harmonic and sense preserving in \({\mathbb{D}}\). We let \(\mathcal{H}\) be the class of functions \(f=h+\overline{g}\), harmonic, sense preserving, and univalent in the open unit disk \({\mathbb{D}}\), for which \(f_{\overline {z}}(0)=g^{\prime }(0)=0\). Such harmonic and sense-preserving functions \(f=h+\overline {g}\in \mathcal{H}\) may be represented by the power series
$$ h(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k}, \qquad g(z)=\sum_{k=2}^{\infty}b_{k}z^{k},\quad z\in{\mathbb{D}}. $$
(1)
Clunie and Sheil-Small [5] proved that the class \(\mathcal{H}\) is a compact family (with respect to the topology of locally uniform convergence). Note that for \(g(z)\equiv0\), the class \(\mathcal{H}\) reduces to the class \(\mathcal{S}\) of normalized analytic functions univalent in \(\mathbb{D}\).
Anzeige
For \(0\leq\alpha<1\) we let \(\mathcal{S}_{\mathcal{H}}^{\ast}(\alpha)\) and \(\mathcal{S}_{\mathcal{H}}^{c}(\alpha)\), respectively, denote the subclasses of \(\mathcal{S}_{\mathcal{H}}\) consisting of harmonic starlike and harmonic convex functions of order α.
A functions f of the form (1) is in \(\mathcal{S}_{\mathcal{H}}^{\ast}(\alpha)\) if and only if (e.g. see Clunie and Sheil-Small [5] or Duren [3])
$$ \frac{\partial}{\partial\theta} \bigl( \arg f \bigl( re^{i\theta } \bigr) \bigr) >\alpha,\quad |z|=r< 1. $$
Similarly, a function f of the form (1) is in \(\mathcal{S}_{\mathcal {H}}^{c}(\alpha)\) if and only if
$$ \frac{\partial}{\partial\theta} \biggl( \arg\frac{\partial}{\partial \theta}f \bigl( \rho e^{i\theta} \bigr) \biggr) >\alpha,\quad |z|=r< 1. $$
We note that a harmonic function \(f\in\mathcal{S}_{\mathcal{H}}^{\ast }(\alpha)\) if and only if or where
$$ \operatorname{Re} \frac{J_{\mathcal{H}}f ( z ) }{f ( z ) }>\alpha,\quad |z|=r< 1, $$
$$ \biggl\vert {\frac{J_{\mathcal{H}}{f}(z)- ( 1+\alpha ) f(z)}{J_{\mathcal{H}}{f}(z)+ ( 1-\alpha ) f(z)}}\biggr\vert < 1,\quad |z|=r< 1, $$
$$ J_{\mathcal{H}}f ( z ) :=zh^{\prime} ( z ) -\overline {zg^{\prime} ( z ) }. $$
Anzeige
For \(0\leq\alpha<1\), it is easy to verify that For \(\lambda\in \{ 0,1,2,\ldots \} \) and \(f=h+\overline {g}\in \mathcal{H}\) of the form (1), we consider the linear operator \(J_{\mathcal{H}}^{\lambda}:\mathcal{H}\rightarrow\mathcal{H}\) defined by \(J_{\mathcal {H}}^{0}f:=f=h+\overline{g}\) and For the analytic definition of the above case, see the Sălăgean operator [6].
$$ f\in\mathcal{S}_{\mathcal{H}}^{c}(\alpha)\quad\Leftrightarrow\quad J_{\mathcal {H}}f\in\mathcal{S}_{\mathcal{H}}^{\ast}(\alpha). $$
$$\begin{aligned} J_{\mathcal{H}}^{\lambda}f(z)& :=J_{\mathcal{H}} \bigl( J_{\mathcal{H}}^{\lambda-1}f(z) \bigr)\\ & :=z+\sum_{n=2}^{\infty}n^{\lambda}a_{n}z^{n}+ ( -1 ) ^{\lambda}\sum_{n=2}^{\infty}n^{\lambda} \overline {b}_{n}\overline{z}^{n} \quad( z\in\mathbb{D} ) . \end{aligned}$$
We say that a function \(f:{\mathbb{D\rightarrow C}}\) is subordinate to a function \(g:{\mathbb{D\rightarrow C}}\), and write \(f(z)\prec g(z)\) (or simply \(f\prec g\)), if there exists a complex-valued function w which maps \({\mathbb{D}}\) into itself with \(w(0)=0\), such that \(f(z)=g(w(z))\); \(z\in{\mathbb{D}}\). In particular, if g is univalent in \({\mathbb {D}}\), then \(f(0)=g(0)\) and \(f({\mathbb{D}})\subset g({\mathbb{D}})\).
The Hadamard product (or convolution) of functions \(f_{1}\) and \(f_{2}\) of the form is defined by
$$ f_{k}(z)=z+\sum_{n=2}^{\infty}a_{k,n}z^{n}+ \sum_{n=2}^{\infty }\overline{b_{k,n}z^{n}}\quad \bigl( z\in{\mathbb{D}}, k\in\{1,2\} \bigr) $$
(2)
$$ ( f_{1}\ast f_{2} ) ( z ) =z+\sum _{n=2}^{\infty }a_{1,n}a_{2,n}z^{n}+ \sum_{n=2}^{\infty}\overline {b_{1,n}b_{2,n}z^{n}}\quad ( z\in{\mathbb{D}} ) . $$
For nonnegative integer \(\lambda\in \{ 0,1,2,\ldots \} \) and for \({-B\leq A< B\leq1}\) we define \(\mathcal{H}^{\lambda}(A,B) \) to be the class of functions \(f\in\mathcal{H}\) so that (also see Dziok [7, 8]) and \(\mathcal{G}^{\lambda}(A,B) \) to consist of functions \(f\in \mathcal{H}\) so that
$$ \frac{J_{\mathcal{H}}^{\lambda+1}f ( z ) }{J_{\mathcal{H}}^{\lambda}f ( z ) }\prec\frac{1+Az}{1+Bz} $$
(3)
$$ \frac{J_{\mathcal{H}}^{\lambda}f ( z ) }{z}\prec\frac{1+Az}{1+Bz}. $$
We remark that the classes \(\mathcal{H}^{0}(A,B)\) and \(\mathcal {G}^{0}(A,B)\) for the analytic case, i.e.
\(g\equiv0\), were introduced by Janowski [9] and the classes \(\mathcal{S}_{\mathcal{H}}^{\ast}(\alpha)=\mathcal {H}^{0}(2\alpha-1,1)\) and \(\mathcal{S}_{\mathcal{H}}^{c}(\alpha)=\mathcal {H}^{1}(2\alpha-1,1)\) for the harmonic case were investigated by Jahangiri [10, 11] and Silverman [12]. It is the aim of this paper to obtain necessary and sufficient convolution conditions, coefficient bounds, distortion theorems, radii of starlikeness and convexity, compactness, and extreme points for the above defined classes \(\mathcal{H}^{\lambda}(A,B)\) and \(\mathcal{G}^{\lambda}(A,B)\).
2 Analytic criteria
Our first theorem provides a necessary and sufficient convolution condition for the harmonic functions in \(\mathcal{H}^{\lambda}(A,B)\).
Theorem 1
A function
f
belongs to the class
\(\mathcal{H}^{\lambda }(A,B)\)
if and only if
\(f\in\mathcal{H}\)
and
where
$$ J_{\mathcal{H}}^{\lambda}f ( z ) \ast\phi ( z;\zeta ) \neq0 \quad\bigl( \zeta \in\mathbb{C}, \vert \zeta \vert =1 \bigr), $$
$$ \phi ( z;\zeta ) =\frac{ ( B-A ) \zeta z+ ( 1+A\zeta ) z^{2}}{ ( 1-z ) ^{2}}- ( -1 ) ^{\lambda}\frac {2\overline{z}+ ( A+B ) \zeta\overline{z}- ( 1+A\zeta ) \overline{z}^{2}}{ ( 1-\overline{z} ) ^{2}} \quad( z \in\mathbb{D}) . $$
Proof
Let \(f\in\mathcal{H}\). Then \(f\in\mathcal{H}^{\lambda}(A,B)\) if and only if the condition (3) holds or equivalently Now for \(J_{\mathcal{H}}^{\lambda+1}h ( z ) =J_{\mathcal{H}}^{\lambda}h ( z ) \ast z/(1-z)^{2}\) and \(J_{\mathcal {H}}^{\lambda }h ( z ) =J_{\mathcal{H}}^{\lambda}h ( z ) \ast z/(1-z)\), the inequality (4) yields □
$$ \frac{J_{\mathcal{H}}^{\lambda+1}f ( z ) }{J_{\mathcal{H}}^{\lambda}f ( z ) }\neq\frac{1+A\zeta}{1+B\zeta} \quad \bigl( \zeta\in\mathbb{C}, \vert \zeta \vert =1 \bigr) . $$
(4)
$$\begin{aligned} & ( 1+B\zeta ) J_{\mathcal{H}}^{\lambda+1}f ( z ) - ( 1+A\zeta ) J_{\mathcal{H}}^{\lambda}f ( z ) \\ &\quad= ( 1+B\zeta ) J_{\mathcal{H}}^{\lambda+1}h ( z ) - ( 1+A\zeta ) J_{\mathcal{H}}^{\lambda}h ( z ) \\ &\qquad{}- ( -1 ) ^{\lambda} \bigl[ ( 1+B\zeta ) J_{\mathcal{H}}^{\lambda+1} \overline{g ( z ) }+ ( 1+A\zeta ) J_{\mathcal{H}}^{\lambda}\overline{g ( z ) } \bigr] \\ &\quad=J_{\mathcal{H}}^{\lambda}h ( z ) \ast \biggl( \frac{ ( 1+B\zeta ) z}{ ( 1-z ) ^{2}}- \frac{ ( 1+A\zeta ) z}{1-z} \biggr) \\ &\qquad{}- ( -1 ) ^{\lambda+1}J_{\mathcal{H}}^{\lambda}\overline {g ( z ) } \ast \biggl( \frac{ ( 1+B\zeta ) \overline{z}}{ ( 1-\overline{z} ) ^{2}}+\frac{ ( 1+A\zeta ) \overline{z}}{1- \overline{z}} \biggr) \\ &\quad=J_{\mathcal{H}}^{\lambda}f ( z ) \ast\phi ( z;\zeta ) \neq0. \end{aligned}$$
A sufficient coefficient bound for the functions in \(\mathcal {H}^{\lambda }(A,B)\) is provided in the following.
Theorem 2
For
\(z\in{\mathbb{D}}\), the harmonic function
\(f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}+\sum_{n=2}^{\infty} \overline{b_{n}z^{n}} \)
is in
\(\mathcal{H}^{\lambda}(A,B)\)
if
where
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{n}\vert +\delta_{n}\vert b_{n} \vert \bigr) \leq B-A, $$
(5)
$$ \gamma_{n}=n^{\lambda} \bigl( n ( 1+B ) - ( 1+A ) \bigr) \quad \textit{and} \quad\delta_{n}=n^{\lambda} \bigl( n ( 1+B ) + ( 1+A ) \bigr) . $$
(6)
Proof
Clearly the theorem is true for \(f ( z ) \equiv z\). So, we assume that \(a_{n}\neq0\) or \(b_{n}\neq0\) for \(n\geq2\). Since \(\gamma _{n}\geq n(B-A)\) and \(\delta_{n}\geq n(B-A)\) by (5) we have Therefore f is sense preserving and locally univalent in \({\mathbb{D}}\). Further, if \(z_{1},z_{2}\in{\mathbb{D}}\) and we assume that \(z_{1}\neq z_{2} \), then Hence This proves that f is univalent in \({\mathbb{D}}\)
i.e.
\(f\in \mathcal{H}\).
$$\begin{aligned} \bigl\vert h^{\prime} ( z ) \bigr\vert -\bigl\vert g^{\prime } ( z ) \bigr\vert & \geq1-\sum_{n=2}^{\infty}n \vert a_{n}\vert \vert z\vert ^{n}-\sum _{n=2}^{\infty }n\vert b_{n}\vert \vert z \vert ^{n}\geq1-\vert z\vert \sum_{n=2}^{\infty} \bigl( n\vert a_{n}\vert +n\vert b_{n}\vert \bigr) \\ & \geq1-\frac{\vert z\vert }{B-A}\sum_{n=2}^{\infty } \bigl( \gamma_{n}\vert a_{n}\vert +\delta_{n} \vert b_{n}\vert \bigr) \geq1-\vert z\vert >0 \quad( z \in\mathbb{D}) . \end{aligned}$$
$$ \biggl\vert \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}}\biggr\vert = \Biggl\vert \sum _{m=1}^{n}z_{1}^{m-1}z_{2}^{n-m} \Biggr\vert \leq \sum_{m=1}^{n}\vert z_{1}\vert ^{m-1}\vert z_{2}\vert ^{n-m}< n\quad (n=2,3,\ldots ) . $$
$$\begin{aligned} \bigl\vert f ( z_{1} ) -f ( z_{2} ) \bigr\vert & \geq \bigl\vert h ( z_{1} ) -h ( z_{2} ) \bigr\vert -\bigl\vert g ( z_{1} ) -g ( z_{2} ) \bigr\vert \\ & \geq\Biggl\vert z_{1}-z_{2}-\sum _{n=2}^{\infty}a_{n} \bigl( z_{1}^{n}-z_{2}^{n} \bigr) \Biggr\vert -\Biggl\vert \sum_{n=2}^{\infty } \overline{b_{n} \bigl( z_{1}^{n}-z_{2}^{n} \bigr) }\Biggr\vert \\ & \geq \vert z_{1}-z_{2}\vert -\sum _{n=2}^{\infty }\vert a_{n}\vert \bigl\vert z_{1}^{n}-z_{2}^{n}\bigr\vert -\sum_{n=2}^{\infty} \vert b_{n} \vert \bigl\vert z_{1}^{n}-z_{2}^{n} \bigr\vert \\ & =\vert z_{1}-z_{2}\vert \Biggl( 1-\sum _{n=2}^{\infty }\vert a_{n}\vert \biggl\vert \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}}\biggr\vert -\sum_{n=2}^{\infty} \vert b_{n}\vert \biggl\vert \frac{z_{1}^{n}-z_{2}^{n}}{z_{1}-z_{2}}\biggr\vert \Biggr) \\ & >\vert z_{1}-z_{2}\vert \Biggl( 1-\sum _{n=2}^{\infty }n\vert a_{n}\vert -\sum _{n=2}^{\infty}n \vert b_{n}\vert \Biggr) \geq0. \end{aligned}$$
On the other hand, \(f\in\mathcal{H}^{\lambda}(A,B)\) if and only if there exists a complex-valued function w, \(w(0)=0\), \(\vert w(z)\vert <1\) (\(z\in\mathbb{D}\)) such that or equivalently The above inequality (7) holds since for \(\vert z\vert =r \) (\(0< r<1\)) we obtain and therefore \(f\in\mathcal{H}^{\lambda}(A,B)\). □
$$ \frac{J_{\mathcal{H}}^{\lambda+1}f ( z ) }{J_{\mathcal{H}}^{\lambda}f ( z ) }=\frac{1+Aw(z)}{1+Bw(z)}\quad ( z\in {\mathbb{D}} ) , $$
$$ \biggl\vert \frac{J_{\mathcal{H}}^{\lambda+1}f ( z ) -J_{\mathcal{H}}^{\lambda}f ( z ) }{BD_{\mathcal{H}}^{\lambda+1}f ( z ) -AD_{\mathcal{H}}^{\lambda}f ( z ) }\biggr\vert < 1 \quad( z\in{\mathbb{D}} ) . $$
(7)
$$\begin{aligned} &\bigl\vert J_{\mathcal{H}}^{\lambda+1}f ( z ) -J_{\mathcal{H}}^{\lambda}f ( z ) \bigr\vert -\bigl\vert BD_{\mathcal{H}}^{\lambda+1}f ( z ) -AD_{\mathcal{H}}^{\lambda}f ( z ) \bigr\vert \\ &\quad =\Biggl\vert \sum_{n=2}^{\infty}n^{\lambda} ( n-1 ) a_{n}z^{n}- ( -1 ) ^{\lambda}\sum _{n=2}^{\infty }n^{\lambda } ( n+1 ) \overline{b}_{n}\overline{z}^{n}\Biggr\vert \\ & \qquad{}-\Biggl\vert ( B-A ) z+\sum_{n=2}^{\infty}n^{\lambda } ( Bn-A ) a_{n}z^{n}+ ( -1 ) ^{\lambda }\sum _{n=2}^{\infty}n^{\lambda} ( Bn+A ) \overline {b}_{n}\overline{z}^{n}\Biggr\vert \\ & \quad\leq\sum_{n=2}^{\infty}n^{\lambda} ( n-1 ) \vert a_{n}\vert r^{n}+\sum _{n=2}^{\infty}n^{\lambda} ( n+1 ) \vert b_{n}\vert r^{n}- ( B-A ) r \\ &\qquad{} +\sum_{n=2}^{\infty}n^{\lambda} ( Bn-A ) \vert a_{n}\vert r^{n}+\sum _{n=2}^{\infty}n^{\lambda} ( Bn+A ) \vert b_{n}\vert r^{n} \\ &\quad \leq r \Biggl\{ \sum_{n=2}^{\infty} \bigl( \gamma_{n}\vert a_{n}\vert +\delta_{n}\vert b_{n}\vert \bigr) r^{n-1}- ( B-A ) \Biggr\} < 0, \end{aligned}$$
Next we show that the condition (5) is also necessary for the functions \(f\in\mathcal{H}\) to be in the class \(\mathcal{H}_{\mathcal {T}}^{\lambda}(A,B):=\mathcal{T}^{\lambda}\cap\mathcal{H}^{\lambda}(A,B)\) where \(\mathcal{T}^{\lambda}\) is the class of functions \(f=h+\overline {g}\in\mathcal{H}\) with varying coefficients (see [13, 14] or [15]) for which there exists a real number ϕ so that
$$ f=h+\overline{g}=z-\sum_{n=2}^{\infty}e^{i ( 1-n ) \phi } \vert a_{n}\vert z^{n}+ ( -1 ) ^{\lambda }\sum _{n=2}^{\infty}e^{i ( n-1 ) \phi} \vert b_{n}\vert \overline{z}^{n} \quad( z\in{\mathbb{D}} ) . $$
(8)
Theorem 3
Proof
The ‘if’ part follows from Theorem 2. For the ‘only-if’ part, assume that \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\), then by (7) we have
$$ \biggl\vert \frac{\sum_{n=2}^{\infty}n^{\lambda} [ ( n-1 ) a_{n}z^{n-1}+ ( -1 ) ^{\lambda} ( n+1 ) \overline{b}_{n}\overline{z}^{n-1} ] }{B-A-\sum_{n=2}^{\infty }n^{\lambda} [ ( Bn-A ) a_{n}z^{n-1}+ ( -1 ) ^{\lambda} ( Bn+A ) \overline{b}_{n}\overline{z}^{n-1} ] }\biggr\vert < 1 \quad(z\in{\mathbb{D}}). $$
For \(|z|=r<1\) we obtain Thus, for \(\gamma_{n}\) and \(\delta_{n}\) as defined by (6), we have Let \(\{ \sigma_{n} \} \) be the sequence of partial sums of the series \(\sum_{n=2}^{\infty} ( \gamma_{n}\vert a_{n}\vert +\delta_{n}\vert b_{n}\vert ) \). Then \(\{ \sigma_{n} \} \) is a nondecreasing sequence and by (9) it is bounded above by \(B-A\). Thus, it is convergent and This gives the condition (5). □
$$ \frac{\sum_{n=2}^{\infty}n^{\lambda} [ ( n-1 ) \vert a_{n}\vert + ( n+1 ) \vert b_{n}\vert ] r^{n-1}}{ ( B-A ) -\sum_{n=2}^{\infty }n^{\lambda} [ ( Bn-A ) \vert a_{n}\vert + ( Bn+A ) \vert b_{n}\vert ] r^{n-1}}< 1. $$
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{n}\vert +\delta_{n}\vert b_{n} \vert \bigr) r^{n-1}< B-A \quad(0\leq r< 1). $$
(9)
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{n}\vert +\delta_{n}\vert b_{n} \vert \bigr) =\lim_{n\rightarrow \infty }\sigma_{n}\leq B-A. $$
A similar argument can be used to prove the following.
Theorem 4
Let
\(f=h+\overline{g}\in\mathcal{H}\)
be a function of the form (8). Then
\(f\in\mathcal{G}_{\mathcal{T}}^{\lambda }(A,B):=\mathcal{T}^{\lambda}\cap\mathcal{G}^{\lambda}(A,B)\)
if and only if
$$ \sum_{n=2}^{\infty}n^{\lambda} \bigl( \vert a_{n}\vert +\vert b_{n}\vert \bigr) \leq \frac{B-A}{1+B}. $$
Corollary 1
Let
\(f=h+\overline{g}\in\mathcal{H}\). Then
\(f\in\mathcal{H}^{\lambda }:=\mathcal{H}^{\lambda}(-1,1)\)
if and only if
where
$$ J_{\mathcal{H}}^{\lambda}f ( z ) \ast\phi ( z;\zeta ) \neq0\quad \bigl( \vert \zeta \vert =1 \bigr) , $$
$$ \phi ( z;\zeta ) =\frac{2\zeta z+ ( 1-\zeta ) z^{2}}{ ( 1-z ) ^{2}}- ( -1 ) ^{\lambda} \frac{2\overline{z}+ ( 1-\zeta ) \overline{z}^{2}}{ ( 1-\overline{z} ) ^{2}}\quad ( z\in{\mathbb{D}} ) . $$
Corollary 2
Let
\(f=h+\overline{g}\in\mathcal{H}\)
be a function of the form (8). Then
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(\alpha):=\mathcal{H}_{\mathcal{T}}^{\lambda}(2\alpha-1,1)\)
if and only if
$$ \sum_{n=2}^{\infty}n^{\lambda} \bigl( ( n- \alpha ) \vert a_{n}\vert + ( n+\alpha ) \vert b_{n} \vert \bigr) \leq1. $$
Corollary 3
Let
\(f=h+\overline{g}\in\mathcal{H}\)
be a function of the form (8). Then
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}:=\mathcal {H}_{\mathcal{T}}^{\lambda}(0)\)
if and only if
i.e.
$$ \sum_{n=2}^{\infty}n^{\lambda+1} \bigl( \vert a_{n}\vert +\vert b_{n}\vert \bigr) \leq1, $$
$$ \mathcal{H}_{\mathcal{T}}^{\lambda}\equiv\mathcal{G}_{\mathcal{T}}^{\lambda+1}:= \mathcal{G}_{\mathcal{T}}^{\lambda+1}(-1,1). $$
3 Extreme points
A function \(f\in\mathcal{F}\subset \mathcal{H}\) is called an extreme point of
\(\mathcal{F}\) if \(f=\mu f_{1}+ ( 1-\mu ) f_{2}\) implies \(f_{1}=f_{2}=f\) for all \(f_{1}\) and \(f_{2}\) in \(\mathcal{F}\) and \(0<\mu<1\). We shall use the notation \(E\mathcal{F}\) to denote the set of all extreme points of \(\mathcal{F}\). It is clear that \(E\mathcal{F}\subset\mathcal{F}\).
We say that \(\mathcal{F}\) is locally uniformly bounded if for each r, \(0< r<1\), there is a real constant \(M=M ( r ) \) so that \(\vert f(z)\vert \leq M\) where \(f\in\mathcal{F}\) and \(\vert z\vert \leq r\).
We say that a class \(\mathcal{F}\) is convex if \(\mu f+(1-\mu )g\in \mathcal{F}\) for all f and g in \(\mathcal{F}\) and \(0\leq\mu\leq 1\). Moreover, we define the closed convex hull of \(\mathcal{F}\), denoted by \(\overline{co}\mathcal{F}\), as the intersection of all closed convex subsets of \(\mathcal{H}\) (with respect to the topology of locally uniform convergence) that contain \(\mathcal{F}\).
A real-valued functional \(\mathcal{J}:\mathcal{H}\rightarrow \mathbb{R} \) is called convex on a convex class \(\mathcal{F}\subset \mathcal{H}\) if \(\mathcal{J} ( \mu f+ ( 1-\mu ) g ) \leq\mu \mathcal{J} ( f ) + ( 1-\mu ) \mathcal{J} ( g ) \) for all f and g in \(\mathcal{F}\) and \(0\leq\mu\leq1\).
The Krein-Milman theorem (see [16]) is fundamental in the theory of extreme points. In particular, it implies the following.
Lemma 1
If
\(\mathcal{F}\)
is a non-empty compact subclass of the class
\(\mathcal{H}\), then
\(E\mathcal{F}\)
is non-empty and
\(\overline {co}E\mathcal{F}=\overline{co}\mathcal{F}\).
Lemma 2
[7]
Let
\(\mathcal{F}\)
be a non-empty compact convex subclass of the class
\(\mathcal{H}\)
and
\(\mathcal{J}:\mathcal{H}\rightarrow \mathbb{R}\)
be a real-valued, continuous, and convex functional on
\(\mathcal{F}\). Then
$$ \max \bigl\{ \mathcal{J}(f):f\in\mathcal{F} \bigr\} =\max \bigl\{ \mathcal{J}(f):f\in E\mathcal{F} \bigr\} . $$
Since \(\mathcal{H}\) is a complete metric space, Montel’s theorem [17] implies the following.
Lemma 3
A class
\(\mathcal{F}\subset\mathcal{H}\)
is compact if and only if
\(\mathcal{F}\)
is closed and locally uniformly bounded.
Now, we are ready to state and prove our next theorem.
Theorem 5
The class
\(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
is a convex and compact subset of
\(\mathcal{H}\).
Proof
For \({0\leq\mu\leq1}\), let \(f_{1},f_{2}\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) be defined by (2). Then and Thus, the function \({\phi}=\mu f_{1}+(1-\mu)f_{2}\) belongs to the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\). This means that the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is convex.
$$ \mu f_{1}(z)+(1-\mu)f_{2} ( z ) =z+\sum _{n=2}^{\infty } \bigl\{ \bigl( \mu a_{1,n}+ ( 1- \mu ) a_{2,n} \bigr) z^{n}+\overline{ \bigl( \mu b_{1,n}+ ( 1-\mu ) b_{2,n} \bigr) z^{n}} \bigr\} $$
$$\begin{aligned} &\sum_{n=2}^{\infty} \bigl\{ \gamma_{n}\bigl\vert \mu a_{1,n}+ ( 1-\mu ) a_{2,n} \bigr\vert +\delta_{n}\bigl\vert \mu b_{1,n}+ ( 1-\mu ) b_{2,n}z^{n}\bigr\vert \bigr\} \\ &\quad\leq\mu\sum_{n=2}^{\infty} \bigl\{ \gamma_{n}\vert a_{1,n}\vert +\delta_{n}\vert b_{1,n}\vert \bigr\} + ( 1-\mu ) \sum_{n=2}^{\infty} \gamma_{n}\vert a_{2,n}\vert +\delta_{n}\vert b_{2,n}\vert \\ &\quad\leq\mu ( B-A ) + ( 1-\mu ) ( B-A ) =B-A. \end{aligned}$$
On the other hand, for \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda }(A,B)\), \(\vert z\vert \leq r\) and \(0< r<1\), we have Therefore, \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is locally uniformly bounded. Let and let \(f=h+\overline{g}\) be given by (1). Using Theorem 3 we have If we assume that \(f_{k}\rightarrow f\), then we conclude that \(\vert a_{k,n}\vert \rightarrow \vert a_{n}\vert \) and \(\vert b_{k,n}\vert \rightarrow \vert b_{n}\vert \) as \(k\rightarrow\infty \) (\(n\in\mathbb{N}\)). Let \(\{ \sigma_{n} \} \) be the sequence of partial sums of the series \(\sum_{n=2}^{\infty} ( \gamma_{n}\vert a_{n}\vert +\delta_{n}\vert b_{n}\vert ) \). Then \(\{ \sigma _{n} \} \) is a nondecreasing sequence and by (10) it is bounded above by \(B-A\). Thus, it is convergent and Therefore, \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\), and therefore the class \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) is closed. In consequence, by Lemma 3, the class \(\mathcal{H}_{\mathcal{T}}^{\lambda }(A,B)\) is compact subset of \(\mathcal{H}\), which completes the proof. □
$$ \bigl\vert f(z)\bigr\vert \leq r+\sum_{n=2}^{\infty} \bigl( \vert a_{n}\vert +\vert b_{n}\vert \bigr) r^{n}\leq r+\sum_{n=2}^{\infty} \bigl( \gamma_{n}\vert a_{n}\vert +\delta_{n} \vert b_{n}\vert \bigr) \leq r+ ( B-A ) . $$
$$ f_{k}(z)=z+\sum_{n=2}^{\infty}a_{k,n}z^{n}+ \sum_{n=2}^{\infty }\overline{b_{k,n}z^{n}}\quad ( z\in{\mathbb{D}}, k\in\mathbb{N}) $$
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{k,n}\vert +\delta_{n}\vert b_{k,n} \vert \bigr) \leq B-A \quad( k\in \mathbb{N}) . $$
(10)
$$ \sum_{n=2}^{\infty} \bigl( \gamma_{n} \vert a_{n}\vert +\delta_{n}\vert b_{n} \vert \bigr) =\lim_{n\rightarrow \infty }\sigma_{n}\leq B-A. $$
Our next theorem is on the extreme points of \(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\).
Theorem 6
Extreme points of the class
\(\mathcal{H}_{\mathcal {T}}^{\lambda } ( A,B ) \)
are the functions
f
of the form (1) where
\(h=h_{n} \)
and
\(g=g_{n} \)
are of the form
$$\begin{aligned} &h_{1}(z) =z,\qquad h_{n}(z)=z-\frac{B-A}{\gamma_{n}}e^{i ( 1-n )\phi }z^{n},\\ &g_{n}(z) = ( -1 ) ^{\lambda}\frac{B-A}{\delta_{n}}e^{i ( n-1 ) \phi } \overline{z}^{n} \quad\bigl(z\in{\mathbb{D}}, n\in \{ 2,3,\ldots \} \bigr). \end{aligned}$$
(11)
Proof
Let \(g_{n}=\mu f_{1}+ ( 1-\mu ) f_{2}\) where \(0<\mu<1\) and \(f_{1},f_{2}\in\mathcal{S}_{\mathcal{T}}^{\lambda} ( A,B ) \) are functions of the form (2). Then, by (5), we have \(\vert b_{1,n}\vert =\vert b_{2,n}\vert =\frac{B-A}{\delta _{n}}\), and therefore \(a_{1,k}=a_{2,k}=0\) for \(k\in \{ 2,3,\ldots \} \) and \(b_{1,k}=b_{2,k}=0\) for \(k\in \{ 2,3,\ldots \} \diagdown \{ n \} \). It follows that \(g_{n}=f_{1}=f_{2}\) and consequently \(g_{n}\in E\mathcal{S}_{\mathcal{T}}^{\ast}(A,B)\). Similarly, we can verify that the functions \(h_{n}\) of the form (11) are the extreme points of the class \(\mathcal{S}_{\mathcal{T}}^{\lambda} ( A,B ) \).
Now, suppose that a function f of the form (1) belongs to the set \(E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) and f is not of the form (11). Then there exists \(m\in \{ 2,3,\ldots \} \) such that If \(0<\vert a_{m}\vert <\frac{B-A}{{\alpha}_{m}}\), then putting we have \(0<\mu<1\), \(h_{m}\neq\phi\), and Thus, \(f\notin E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\). Similarly, if \(0<\vert b_{m}\vert <\frac{B-A}{\delta_{n}}\), then putting we have \(0<\mu<1\), \(g_{m}\neq\phi\), and It follows that \(f\notin E\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\), and so the proof is completed. □
$$ 0< \vert a_{m}\vert < \frac{B-A}{{\alpha}_{m}}\quad\mbox{or}\quad 0< \vert b_{m}\vert < \frac{B-A}{{\beta}_{m}}. $$
$$ \mu=\frac{\vert a_{m}\vert {\alpha}_{m}}{B-A}, \qquad\phi=\frac {1}{1-\mu} ( f-\mu h_{m} ) , $$
$$ f=\mu h_{m}+ ( 1-\mu ) \phi. $$
$$ \mu=\frac{\vert b_{m}\vert {\beta}_{m}}{B-A},\qquad \phi=\frac {1}{1-\mu} ( f-\mu g_{m} ) , $$
$$ f=\mu g_{m}+ ( 1-\mu ) \phi. $$
It is clear that if the class \(\mathcal{F}= \{ f_{n}\in\mathcal {H}: n\in\mathbb{N}\} \) is locally uniformly bounded, then Thus, by Theorem 6, we have the following.
$$ \overline{co}\mathcal{F}= \Biggl\{ \sum_{n=1}^{\infty} \mu _{n}f_{n}: \sum_{n=1}^{\infty} \mu_{n}=1, \mu_{n}\geq0\ ( n\in \mathbb{N}) \Biggr\} . $$
Corollary 4
Let
\(h_{n}\), \(g_{n}\)
be defined by (11). Then
$$ \mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)= \Biggl\{ \sum _{n=1}^{\infty } ( \mu_{n}h_{n}+ \delta_{n}g_{n} ) : \sum_{n=1}^{\infty} ( \mu _{n}+\delta_{n} ) =1, \delta_{1}=0, \mu_{n},\delta_{n}\geq 0\ ( n\in\mathbb{N} ) \Biggr\} . $$
For all fixed values of \(m,n,\lambda\in\mathbb{N}\), \(z\in{\mathbb{D}}\), the following real-valued functionals are continuous and convex on \(\mathcal{H}\): Moreover, for \(\mu>0\), \(0< r<1\), the real-valued functional is continuous on \(\mathcal{H}\). For \(\mu\geq1\), by Minkowski’s inequality it is also convex on \(\mathcal{H}\).
$$ \mathcal{J} ( f ) =\vert a_{n}\vert , \qquad \mathcal{J} ( f ) =\vert b_{n}\vert , \qquad\mathcal {J} ( f ) =\bigl\vert f ( z ) \bigr\vert , \qquad\mathcal{J} (f ) =\bigl\vert J_{\mathcal{H}}^{\lambda}f ( z ) \bigr\vert \quad ( f\in{ \mathcal{H}} ) . $$
$$ \mathcal{J} ( f ) = \biggl( \frac{1}{2\pi} \int_{0}^{2\pi }\bigl\vert f \bigl( re^{i\theta} \bigr) \bigr\vert ^{\mu}d\theta \biggr) ^{1/\mu} \quad( f\in{ \mathcal{H}} ) $$
Corollary 5
Let
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
be a function of the form (8). Then
where
\(\gamma_{n}\), \(\delta_{n}\)
are defined by (6). The result is sharp and the functions
\(h_{n}\), \(g_{n}\)
of the form (11) are the extremal functions.
$$ \vert a_{n}\vert \leq\frac{B-A}{\gamma_{n}}, \qquad \vert b_{n}\vert \leq\frac{B-A}{\delta_{n}}\quad (n=2,3,\ldots), $$
Corollary 6
Let
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
and
\(\vert z\vert =r<1\). Then
The result is sharp and the function
\(h_{2}\)
of the form (11) is the extremal function.
$$\begin{aligned}& r-\frac{B-A}{2^{\lambda} ( 1+2B-A ) }r^{2} \leq\bigl\vert f(z)\bigr\vert \leq r+ \frac{B-A}{2^{\lambda} ( 1+2B-A ) }r^{2},\\& r-\frac{B-A}{1+2B-A}r^{2} \leq\bigl\vert J_{\mathcal{H}}^{\lambda }f(z) \bigr\vert \leq r+\frac{B-A}{1+2B-A}r^{2} \quad( \lambda =1,2,3,\ldots ) . \end{aligned}$$
The following covering result follows from Corollary
6.
Corollary 7
If
\(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\)
then
\({\mathbb{D}} ( r ) \subset f ( {\mathbb{D}} ) \)
where
We also conclude to the following.
$$ r=1-\frac{B-A}{2^{\lambda} ( 1+2B-A ) }. $$
Corollary 8
Let
\(0< r<1\)
and
\(\mu\geq1\). If
\(f\in\mathcal{H}_{\mathcal {T}}^{\lambda }(A,B)\)
then
$$\begin{aligned}& \frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert f\bigl(re^{i\theta} \bigr)\bigr\vert ^{\mu}d\theta\leq\frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert h_{2} \bigl(re^{i\theta}\bigr)\bigr\vert ^{\mu}d\theta, \\& \frac{1}{2\pi} \int_{0}^{2\pi}\bigl\vert J_{\mathcal {H}}^{\lambda }f(z) \bigr\vert ^{\mu}d\theta\leq\frac{1}{2\pi} \int _{0}^{2\pi }\bigl\vert J_{\mathcal{H}}^{\lambda}h_{2} \bigl(re^{i\theta}\bigr)\bigr\vert ^{\mu }d\theta \quad( \mu=1,2,\ldots ) . \end{aligned}$$
4 Radii of starlikeness and convexity
Let \(\mathcal{B\subseteq H}\) and let \(\mathbb{D}(r): =\{z\in\mathbb{C}: \vert {z}\vert < r\leq1\}\). We define the radius of starlikeness and the radius of convexity of the class \(\mathcal{B}\), respectively, by At this point, for the case \(\alpha=0\), it is easy to verify that and consequently In the following theorem we determine \(R_{\alpha}^{\ast}(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B))\) for \(0\leq\alpha<1\).
$$\begin{aligned} &R_{\alpha}^{\ast}(\mathcal{B}) :=\inf_{f\in\mathcal{B}} \bigl( \sup \bigl\{ r\in(0,1]:f\mbox{ is starlike of order }\alpha\mbox{ in } {\mathbb{D}}(r) \bigr\} \bigr) ,\\ &R_{\alpha}^{c}(\mathcal{B}) :=\inf_{f\in\mathcal{B}} \bigl( \sup \bigl\{ r\in(0,1]:f\mbox{ is convex of order }\alpha\mbox{ in } {\mathbb{D}}(r) \bigr\} \bigr) . \end{aligned}$$
$$ \mathcal{H}^{\lambda}(A,B)\subset\mathcal{H}^{\lambda}(-1,1)\subset \mathcal{H}^{\lambda-1}(-1,1)\subset\mathcal{S}_{\mathcal{H}}^{\ast }(0) \subset\mathcal{H}, $$
$$ R_{0}^{\ast}\bigl(\mathcal{H}^{\lambda}(A,B) \bigr)=R_{0}^{\ast}\bigl(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B) \bigr)=R_{0}^{c}\bigl(\mathcal{H}^{\lambda }(A,B) \bigr)=R_{0}^{c}\bigl(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B) \bigr)=1\quad ( \lambda=2,3,\ldots ) . $$
Theorem 7
Let
\(0\leq\alpha<1\)
and
\(\gamma_{n}\)
and
\(\delta_{n}\)
be defined by (6). Then
$$ {R_{\alpha}^{\ast}\bigl(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B) \bigr)}=\inf_{n\geq2} \biggl( \frac{1-\alpha}{B-A}\min \biggl\{ \frac {\gamma _{n}}{n-\alpha},\frac{\delta_{n}}{n+\alpha} \biggr\} \biggr) ^{{\frac {1}{n-1}}}. $$
(12)
Proof
Let \(f\in\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B)\) be of the form (8). Then, for \(\vert z\vert =r<1\), we have Note (see Jahangiri [11], Theorem 2) that f is starlike of order α in \({\mathbb{D}}(r)\) if and only if or Also, by Theorem 1, we have where \(\gamma_{n}\) and \(\delta_{n}\) are defined by (6).
$$\begin{aligned} \biggl\vert {\frac{J_{\mathcal{H}}{f}(z)- ( 1+\alpha ) f(z)}{J_{\mathcal{H}}{f}(z)+ ( 1-\alpha ) f(z)}}\biggr\vert & =\biggl\vert {\frac{-\alpha z+\sum_{n=2}^{\infty} ( (n-1-\alpha )a_{n}z^{n}-(n+1+\alpha)b_{n}\overline{z}^{n} ) }{ ( 2-\alpha ) z+\sum_{n=2}^{\infty} ( (n+1-\alpha )a_{n}z^{n}-(n-1+\alpha)b_{n}\overline{z}^{n} ) }}\biggr\vert \\ & \leq{\frac{\alpha+\sum_{n=2}^{\infty} ( (n-1-\alpha )\vert a_{n}\vert +(n+1+\alpha)\vert b_{n}\vert ) r^{n-1}}{2-\alpha-\sum_{n=2}^{\infty} ( (n+1-\alpha )\vert a_{n}\vert +(n-1+\alpha)\vert b_{n}\vert ) r^{n-1}}.} \end{aligned}$$
$$ \biggl\vert {\frac{J_{\mathcal{H}}{f}(z)- ( 1+\alpha ) f(z)}{J_{\mathcal{H}}{f}(z)+ ( 1-\alpha ) f(z)}}\biggr\vert < 1 \quad\bigl(z\in{\mathbb{D}}(r)\bigr) $$
$$ \sum_{n=2}^{\infty} \biggl( \frac{n-\alpha}{1-\alpha} \vert a_{n}\vert +\frac{n+\alpha}{1-\alpha} \vert b_{n}\vert \biggr) r^{n-1}\leq1. $$
(13)
$$ \sum_{n=2}^{\infty} \biggl( \frac{\gamma_{n}}{B-A} \vert a_{n}\vert +\frac{\delta_{n}}{B-A}\vert b_{n}\vert \biggr) \leq1, $$
Since \(\gamma_{n}<\delta_{n} \) (\(n=2,3,\ldots\)), the condition (13) is true if or if It follows that the function f is starlike of order α in the disk \(U ( {r}^{\ast} ) \) where The function proves that the radius \(r^{\ast}\) cannot be any larger. Thus we have (12). □
$$ \frac{n-\alpha}{1-\alpha}r^{n-1}\leq\frac{\gamma_{n}}{B-A}\quad\mbox{and}\quad \frac{n+\alpha}{1-\alpha}r^{n-1}\leq\frac{\delta_{n}}{B-A} \quad(n=2,3,\ldots), $$
$$ r\leq \biggl( \frac{1-\alpha}{B-A}\min \biggl\{ \frac{\gamma _{n}}{n-\alpha},\frac{\delta_{n}}{n+\alpha} \biggr\} \biggr) ^{{\frac {1}{n-1}}}\quad(n=2,3,\ldots). $$
$$ { }r^{\ast}:=\inf_{n\geq2} \biggl( \frac{1-\alpha}{B-A} \min \biggl\{ \frac{\gamma_{n}}{n-\alpha},\frac{\delta_{n}}{n+\alpha } \biggr\} \biggr) ^{{\frac{1}{n-1}}}. $$
$$ f_{n}(z)=h_{n}(z)+\overline{g_{n}(z)}=z- \frac{B-A}{\gamma_{n}}e^{i ( 1-n ) \phi}z^{n}+ ( -1 ) ^{\lambda} \frac{B-A}{\delta _{n}}e^{i ( n-1 ) \phi}\overline{z}^{n} $$
Using a similar argument as above we obtain the following.
Theorem 8
Let
\(0\leq\alpha<1\)
and
\(\gamma_{n}\)
and
\(\delta_{n}\)
be defined by (6). Then
$$ {R_{\alpha}^{c}\bigl(\mathcal{H}_{\mathcal{T}}^{\lambda}(A,B) \bigr)}=\inf_{n\geq2} \biggl( \frac{1-\alpha}{B-A}\min \biggl\{ \frac {\gamma _{n}}{n ( n-\alpha ) },\frac{\delta_{n}}{n ( n+\alpha ) } \biggr\} \biggr) ^{{\frac{1}{n-1}}}. $$
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