Abstract
The object of the present paper is to investigate various conditions for Carathéodory functions in the open unit disk. Also we give some applications to univalent functions as special cases.
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1 Introduction
For given r (\(0 < r \leq1\)), let \(\mathbb{U}_{r}=\{ z\in \mathbb{C}: |z|< r \}\), \(\mathbb{U} \equiv\mathbb{U}_{1}\) be the open unit disk, and let us denote by \(\mathbb{T} = \partial\mathbb{U} := \{ z\in\mathbb{C}: |z|=1 \}\) the boundary of \(\mathbb{U}\). An analytic function p in \(\mathbb{U}\) with \(p(0)=1\) is said to be a Carathéodory function of order α if it satisfies
We denote by \(\mathcal{P}({\alpha})\) the class of all Carathéodory functions of order α in \(\mathbb{U}\) and \(\mathcal{P} \equiv \mathcal{P}(0)\) [4]. Let \(\mathcal{A}\) denote the class of analytic functions f defined in \(\mathbb{U}\) normalized by \(f(0)=0\) and \(f'(0)=1\). Further, we denote by \(\mathcal{S}^{*}({\alpha})\) and \(\mathcal{K}({\alpha})\) the subclasses of \(\mathcal{A}\) consisting of starlike and convex functions of order α in \(\mathbb{U}\), respectively. That is, a function \(f \in{\mathcal{A}}\) belongs to the classes \(\mathcal{S}^{*}({\alpha})\) and \(\mathcal{K}({\alpha})\) if f satisfies \(\operatorname {Re}\{zf'(z)/f(z) \} >\alpha\) and \(\operatorname {Re}\{ 1+ zf''(z)/f'(z) \} > \alpha\), respectively, in \(\mathbb{U}\).
For analytic functions f and g, we say that f is subordinate to g, denoted by \(f\prec g\), if there is an analytic function \(w:\mathbb{U}\rightarrow\mathbb{U}\) with \(|w(z)|\leq|z|\) such that \(f(z)=g(w(z))\). Further, if g is univalent, then the definition of subordination \(f\prec g\) simplifies to the conditions \(f(0)=g(0)\) and \(f(\mathbb{U})\subseteq g(\mathbb{U})\) (see [10, p. 36]).
Let us denote by \({\mathcal{Q}}\) the set of functions q that are analytic and injective on \(\overline{\mathbb{U}}\setminus{\mathbf{E}}(q)\), where
and are such that \(q'(\zeta)\neq0\) for \(\zeta\in\mathbb {T}\setminus {\mathbf{E}}(q)\).
Marx [3] and Strohhäcker [12] showed that if \(f\in\mathcal{K} \equiv\mathcal{K}(0)\) then \(f\in\mathcal{S}^{*}( {1/2})\), that is, \(\mathcal{K} \subset\mathcal{S}^{*}({1/2})\). Later, Miller [4] and Miller, Mocanu and Reade [7] proved the following results, respectively. If p is analytic in \(\mathbb{U}\), then
and
The result given in (2) clearly reduces the earlier works due to Marx and Strohhäcker. Many kinds of functions with geometric properties, such as starlikeness, convexity, close-to-convexity, and so on, are closely related to the class of Carathéodory functions and play a really important role in the study of univalent functions.
In the present paper, we show several new sufficient conditions, which are not connected to some recent results for Carathéodory functions of order α, which incorporate the implications given by (1) and (2). In addition to applying the well known Jack’s Lemma, we approach the results in a quite different way than methods used in other papers. Moreover, we obtain other criteria for Carathéodory functions of order α. Many of the earlier results given by Marx [3], Strohhäcker [12] and others are shown here to follow as special cases of the results presented in this paper. Thus the various properties associated with the class \({\mathcal{P}}(\alpha)\) obtained here can be viewed as extensions and generalizations of numerous previously-obtained results in Geometric Function Theory.
2 Main results
In proving our results, we need the following lemmas due to Jack [2], and Miller and Mocanu [5] (see also [6, p. 24, Lemma 2.2d]).
Lemma 2.1
Suppose that function w is analytic for \(|z| \leq r\), \(w(0)=0\) and \(|w(z_{0})| = \max_{|z|=r}|w(z)|\). Then \(z_{0}w^{\prime}(z_{0}) = kw(z _{0})\), where k is a real number with \(k\geq1\).
Lemma 2.2
Let \(q \in{\mathcal{Q}}\), with \(q(0)=a\), and let \(p(z)=a+a_{n} z^{n} + \cdots\) be analytic in \(\mathbb{U}\) with \(p(z)\not\equiv a\) and \(n \geq1\). If p is not subordinate to q, then there exist points \(z_{0} = r_{0} \mathrm {{e}}^{\mathrm {{i}}\theta_{0}} \in\mathbb{U}\) and \(\zeta_{0} \in\mathbb{T}\setminus{\mathbf{E}}(q)\), and an \(m\geq n\geq1\) for which \(p(\mathbb{U}_{r_{0}}) \subset q(\mathbb{U})\),
-
(i)
\(p(z_{0})=q(\zeta_{0})\),
-
(ii)
\(z_{0}p'(z_{0}) = m \zeta_{0} q'(\zeta_{0})\) and
-
(iii)
\(\operatorname {Re}\{ 1 + \frac{z_{0}p''(z_{0})}{p'(z _{0})} \} \geq m \operatorname {Re}\{ 1 + \frac{\zeta_{0} q''(\zeta _{0})}{q'(\zeta_{0})} \}\).
By using Lemma 2.1, we now derive the following theorem.
Theorem 2.3
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
then \(p \in\mathcal{P}({\alpha})\).
Proof
Define function w by
We know that w is analytic in \(\mathbb{U}\) with \(w(0)=0\). Suppose that there exists a point \(z_{0}\) in \(\mathbb{U}\) such that
Then we have
By using Lemma 2.1, we get
where k is a real number with \(k \geq1\). We note that \(z_{0}p'(z _{0})\) is a nonpositive real number, since
and, by (6), \(\operatorname {Re}\{w(z_{0}) \} \leq1\). Moreover, by putting
we obtain
and
Therefore, from equations (9) and (11), we have
This contradicts assumption (3). Therefore we complete the proof of Theorem 2.3. □
Taking \({\alpha}=0\) and \({\beta}=1\) in Theorem 2.3, we have the following result by Nunokawa et al. [9].
Corollary 2.4
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
then \(p\in\mathcal{P}\).
Remark 2.5
Corollary 2.4 is an improvement of the result by Miller [4].
The right-hand side of assumption (3) in Theorem 2.3 depends on \(|p(z)|\). But applying the same method as in the proof of Theorem 2.3 and using the new formula (12) where y is ignored, we can derive a similar result (Theorem 2.6 below) without requiring \(|p(z)|\) in assumption (3) of Theorem 2.3.
Theorem 2.6
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
then \(p \in\mathcal{P}({\alpha})\).
Letting \({\beta}=1\) in Theorem 2.6, we have the following corollary.
Corollary 2.7
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
then \(p \in\mathcal{P}({\alpha})\).
Remark 2.8
Corollary 2.7 is an improvement of the result by Nunokawa [8].
For given γ and c satisfying \(\gamma>0\) and \(c>-\gamma\), let us consider an integral operator \(I_{c,{\gamma}}:{\mathcal{A}} \rightarrow {\mathcal{A}}\) defined by
By taking \(p(z)=F'(z)(F(z)/z)^{\gamma-1}\), we have
Moreover, taking derivatives of both sides of (14) leads to the equality
Example 2.9
Taking \(p(z)=f'(z)\) in Theorem 2.3 with \(\alpha=0\), \(p(z)=f'(z)\) in Theorem 2.6 with \(\alpha=0\) and \(\beta=1\), \(p(z)=f(z)/z\) in Theorem 2.3 with \(\alpha=0\) and \(\beta =1\) and \(p(z)=F'(z)/(F(z)/z)^{\gamma-1}\), where F is defined in (13), in Theorem 2.6 with \(\beta =1/(c+\gamma)\), respectively, we have the following results: If \(f\in\mathcal{A}\), then
-
(i)
\(\operatorname {Re}\{ f^{\prime}(z) + {\beta}zf^{\prime \prime}(z) \} > -\frac{{\beta}}{2}(1+|f^{\prime}(z)|^{2})\) (\({\beta}> 0\)) implies \(\operatorname {Re}\{ f'(z) \} > 0 \) (cf. [1]);
-
(ii)
\(\operatorname {Re}\{ f^{\prime}(z) + zf^{\prime \prime}(z) \} > -1/2\) implies \(\operatorname {Re}\{f'(z) \} > 0\);
-
(iii)
\(\operatorname {Re}\{f^{\prime}(z) \} > - \frac{1}{2} ( 1+ \vert f(z)/z \vert ^{2} )\) implies \(\operatorname {Re}\{ f(z)/z \} > 0\);
-
(iv)
\(\operatorname {Re}\{ f'(z) ( f(z)/z ) ^{{\gamma}-1} \} > {\alpha}- \frac{{1-{\alpha}}}{2(c+{\gamma})}\) (\(0 \leq{\alpha}< 1\), \({\gamma}>0\), \(c > - {\gamma}\)) implies \(\operatorname {Re}\{ F'(z) ( F(z)/z )^{{\gamma}-1} \} > {\alpha}\), where F is defined as in (13) (cf. [11]).
Theorem 2.10
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
where
then \(p\in\mathcal{P}({\alpha})\).
Proof
At first, we note that \(p(z) \neq- {\gamma}/{\beta}\) for \(z\in\mathbb{U}\). In fact, if \({\beta}p(z)+{\gamma}\) has a zero of order m at \(z=z_{1}\in\mathbb{U}\), then we can write
where \(p_{1}\) is analytic in \(\mathbb{U}\) and \(p_{1}(z_{1}) \neq0\). Then we have
Thus choosing \(z \rightarrow z_{1}\) suitably, the real part of the right-hand side of (17) can take any negative infinite values, which contradicts hypothesis (15). Defining w by (4), we see that function w is analytic in \(\mathbb{U}\) with \(w(0)=0\). Suppose that there exists a point \(z_{0} \in\mathbb{U}\) satisfying (5). Then we have (6). By Lemma 2.1, there exists a real number k with \(k\geq1\) satisfying (7). Using the fact that \(z_{0}p'(z_{0})\) is a real number, from (4) and (8), we can obtain
We now set \(p(z_{0})\) as in (9). Then we have the same function value of \(w(z_{0})\) which satisfies formula (10), and it follows from (18) with (10) and \(k\geq1\) that
where \({\delta}({\alpha}, {\beta}, {\gamma},|p(z_{0})|)\) is given by (16), which contradicts assumption (15). Therefore we complete the proof of Theorem 2.10. □
Remark 2.11
For \({\gamma}=0\), Theorem 2.10 is an improvement of the result by Miller et al. [7].
Taking \({\beta}=1\) and \({\gamma}=0\) in Theorem 2.10, we have the following result.
Corollary 2.12
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\) and \(0 \leq{\alpha }<1\). If
then \(p \in\mathcal{P}({\alpha})\).
Applying Theorem 2.10 leads us to get the following theorem which doesn’t depend on \(|p(z)|\).
Theorem 2.13
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\) and \(0 \leq{\alpha }<1\). If p satisfies one of the following conditions:
-
(i)
\(\operatorname {Re}\{ p(z) + \frac{zp^{\prime}(z)}{ {{\beta}p(z)+{\gamma}}} \} > {\alpha}- \frac{{{\alpha} {\beta}+{\gamma}}}{2{\beta}^{2}(1-{\alpha})} \) (\(-{\alpha} {\beta}<{\gamma}<{\beta}(1-2{\alpha})\) for \({\beta}>0\) or \(-{\alpha} {\beta}<{\gamma}<-{\beta}\) for \({\beta}<0\)),
-
(ii)
\(\operatorname {Re}\{ p(z) + \frac{zp^{\prime}(z)}{ {{\beta}p(z)+{\gamma}}} \} > {\alpha}-\frac{{1-{\alpha}}}{2({\alpha} {\beta}+{\gamma})} \) (\({\gamma}\geq {\beta}(1-2{\alpha})\) for \({\beta}>0\) or \({\gamma}\geq-{\beta}\) for \({\beta}<0\)),
then \(p\in\mathcal{P}({\alpha})\).
Proof
First of all, we consider a function \(\psi:[0,\infty) \rightarrow \mathbb{R}\) defined by
By differentiating ψ, we obtain
Therefore the derivative of ψ is negative when \(-{\alpha} {\beta}<{\gamma}<-{\beta}(2{\alpha}-1)\) for \({\beta}>0\) and \(-{\alpha} {\beta}<{\gamma}<-{\beta}\) for \({\beta}<0\), which means that function ψ is decreasing. Hence
On the other hand, the derivative of ψ is positive when \(-{\beta}(2{\alpha}-1)<{\gamma}\) for \({\beta}>0\) and \(-{\beta}<{\gamma}\) for \({\beta}<0\), which means that function ψ is increasing. In this case, the following inequality holds:
According to the same contradiction method as in Theorem 2.10, when \(p(z_{0})\) is defined by (9), we now have
where ψ is the function defined by (19).
When \(-{\alpha} {\beta}<{\gamma}<-{\beta}(2{\alpha}-1)\) for \({\beta}>0\) and \(-{\alpha }{\beta}<{\gamma}<-{\beta}\) for \({\beta}<0\), by (22) and (20), we have
This is a contradiction to the assumption. And, when \(-{\beta}(2{\alpha }-1)<{\gamma}\) for \({\beta}>0\) and \(-{\beta}<{\gamma}\) for \({\beta}<0\), by (22) and (21), we have
But this also contradicts our assumption. Hence the proof of Theorem 2.13 is completed. □
Letting \({\beta}=1\) and \({\gamma}=0\) in Theorem 2.13, we have the following corollary.
Corollary 2.14
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If function p satisfies the following condition, then \(p\in\mathcal{P}({\alpha})\):
Remark 2.15
Taking \(p(z)=zf^{\prime}(z)/f(z)\) and \({\alpha}=1/2\) in Corollary 2.14, we have the classical result by Marx [3] and Strohhäcker [12], that is, \(\mathcal{K} \subset\mathcal{S}^{*}(1/2)\).
Let α and β be real numbers such that \(0 \leq\alpha<1\) and \(\beta\geq(3\alpha-1)/2\). Then it can be easily shown that
Hence it follows that the following inequality holds for \(y\in \mathbb{R}\) and \(k \geq1\):
Now let \(p(z_{0})\) and \(z_{0}p'(z_{0})\) be given as in (9) and (11), respectively. Then, from (23) and replacing \(y^{2}\) by \(|p(z_{0})|^{2}-\alpha^{2}\), we have
Now, applying the same method as in the proof of Theorems 2.3 and 2.10 and inequality (24), we obtain the following result.
Theorem 2.16
Let α and β be real numbers such that \(0 \leq\alpha<1\) and \(\beta\geq(3\alpha-1)/2\). Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
where
then \(p\in\mathcal{P}({\alpha})\).
Corollary 2.17
Let \(0 \leq\alpha<1\) and \(\beta\geq(3\alpha-1)/2\). And let p be an analytic function in \(\mathbb{U}\) with \(p(0)=1\). If p satisfies
then \(p \in{\mathcal{P}}(\alpha)\).
Taking \({\alpha}=0\) and \({\beta}=1\) in Theorem 2.16 and Corollary 2.17, we have Corollary 2.18 below.
Corollary 2.18
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If
or
then \(p\in{\mathcal{P}}\).
With the aid of Lemma 2.2, we prove the following result.
Theorem 2.19
Let α and A be real numbers with \(0 \leq\alpha<1\) and \(A \geq0\). And let B and C be functions defined in \(\mathbb{U}\) such that \(\operatorname {Re}\{ B(z) \} > A\) for all \(z\in\mathbb{U}\). If p is analytic in \(\mathbb{U}\) with \(p(0)=1\) and
where
then \(p \in{\mathcal{P}}(\alpha)\).
Proof
Define function w as in (4). We see that w is analytic in \(\mathbb{U}\) with \(w(0)=0\). Suppose that there exists a point \(z_{0}\) in \(\mathbb{U}\) satisfying (5). Then we have (6). By Lemma 2.1, there exist a real number \(k\geq1\) satisfying (7). Moreover, by hypothesis (5), we have \(p \nprec h\), where \(h:\mathbb{U} \rightarrow\mathbb{C}\) is the function defined by \(h(z)=(1+(1-2 \alpha)z)/(1-z)\). Note that
for \(\zeta\in\mathbb{T}\). Lemma 2.2 with the equality above leads to the inequality
Since \(z_{0}p'(z_{0})\) is a nonpositive real number, from (25), we have
Putting
we obtain the same function value of \(w(z_{0})\) which satisfies equation (5). Then, by (26) and (11), we have the following inequalities:
But this contradicts our assumption. Hence the proof is completed. □
Taking \(A=0\), \(B(z)=C(z)\equiv1\) and \({\alpha}=0\) in Theorem 2.19, then we have the following result.
Corollary 2.20
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). Then
Remark 2.21
Corollary 2.20 is the result obtained by Miller [4]. And this is also shown by Corollary 2.7.
Taking \(A=0\), \(B(z)=C(z)\equiv1\) and \({\alpha}=1/2\) in Theorem 2.19, we have the following result.
Corollary 2.22
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). Then
Letting \(p(z)= ( I_{{\gamma}, {\beta}}[f](z)/z )^{{\beta}}\) (\(f\in \mathcal{A}\)), where \(I_{{\gamma}, {\beta}}: {\mathcal{A}}\rightarrow{\mathcal{A}}\) is the integral operator defined by (13), in Theorem 2.19 with \(A=0\), \(B(z)\equiv 1\), \(C(z)\equiv {\gamma}+{\beta}\) and \({\alpha}=0\), we have the following result.
Corollary 2.23
Let \(f\in\mathcal{A}\) and let β and γ be complex numbers. If
then
By a similar method as in the proof of Theorem 2.19, we can obtain the following result, which shows that the condition \(\operatorname {Re}\{B(z) \} \geq A\) (\(z\in\mathbb{U}\)) can be established in Theorem 2.19 when \(\operatorname {Im}\{ C(z) \}=0\) (\(z\in\mathbb{U}\)).
Theorem 2.24
Let α and A be real numbers with \(0 \leq\alpha<1\) and \(A \geq0\). And let B and C be functions defined in \(\mathbb{U}\) such that \(\operatorname {Re}\{ B(z) \} = A\) and \(\operatorname {Im}\{ C(z) \}=0\) for all \(z\in\mathbb{U}\). If p is analytic in \(\mathbb{U}\) with \(p(0)=1\) and
then \(p \in{\mathcal{P}}(\alpha)\).
Taking \(A=1\), \(B(z)=C(z)\equiv1\) in Theorem 2.19, then we have the following result.
Corollary 2.25
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). Then
Next, we derive another conditions for Carathéodory functions of order α in Theorems 2.26 and 2.27 below.
Theorem 2.26
Let p be analytic in \(\mathbb{U}\) with \(p(0)=1\) and \(0 \leq{\alpha }< 1\). If p satisfies
for all \(\varLambda\in\mathbb{R}\) with \(|\varLambda|\geq1\), then \(p\in\mathcal{P}({\alpha})\).
Proof
Let
Then q is analytic in \(\mathbb{U}\) with \(q(0)=1\). Here, we note that \(p(z) \neq{\alpha}\) for \(z\in\mathbb{U}\). In fact, if there exists a point \(z_{1}\in\mathbb{U}\) such that \(p(z_{1})={\alpha}\) and hence \(q(z_{1})=0\) then \(q(z)\) can written by
where \(q_{1}\) is analytic in \(\mathbb{U}\) and \(q_{1}(z_{1})\neq0 \). Hence we have
But the imaginary part of the right-hand side of (28) can take any value when z approaches \(z_{1}\). This contradicts our assumption (27). Suppose that there exists a point \(z_{0}\in\mathbb{U}\) such that
Setting
we have
Let \(q(z_{0})=\mathrm {{i}}y\) (\(y\in\mathbb{R} \setminus\{0\}\)). Then, by Lemma 2.1, we obtain
where k is a real number with \(k \geq1\), and so
Therefore, \(z_{0}q^{\prime}(z_{0})\) is a negative real number. At first, suppose that \(y>0\). Then we have
Hence we obtain
which contradicts assumption (27). Next, for \(y<0\), we have
and Λ is a real number with \(\varLambda\leq-1\). This also contradicts assumption (27). Hence we complete the proof of Theorem 2.26. □
Theorem 2.27
Let \(0 \leq{\alpha}< 1\) and let p be analytic in \(\mathbb{U}\) with \(p(0)=1\). If p satisfies
for all \(\Delta\in\mathbb{R}\) with \(|\Delta| \geq2\), then \(p\in\mathcal{P}({\alpha})\).
Proof
Firstly, using a proof similar to that of Theorem 2.26 and assumption (29), we can derive easily that
for all \(z\in\mathbb{U}\). Let
Then we see that w is analytic in \(\mathbb{U}\) with \(w(0)=0\). We claim that \(|w(z)|<1\) in \(\mathbb{U}\). Suppose that there exists a point \(z_{0}\in\mathbb{U}\) such that \(\max_{|z|<|z_{0}|} |w(z)| = |w(z_{0})| = 1\). By Lemma 2.1, there exists a real number \(k\geq1\) satisfying (7). Writing \(w(z_{0})=\mathrm {{e}}^{\mathrm {{i}}\theta}\) with
we obtain that
and
Therefore we have the following identities:
It is now sufficient to show that
for all θ satisfying (31), since (32) contradicts the assumption (29). For this, let us define a function \(\varphi:(0,1)\rightarrow\mathbb{R}\) by
We can check \(\varphi'(t)=0\) occurs only at \(t = \sqrt{2}-1 =: t _{0} \in(0,1)\). Moreover, we have \(\varphi''(t_{0})=12+8\sqrt{2}>0\). Therefore, on the interval \((0,1)\), function φ has its minimum at \(t=t_{0}\). That is,
And, by (33), the following inequality holds for \(t \in(1,\infty)\):
Consider the case \(0<\theta<\pi/2\). Then, we have \(\cot(\theta/2)>1\) and it follows from (34) that
For the case \(\pi/2 <\theta<\pi\), we have \(0<\cot(\theta/2)<1\) and (33) gives us that
A similar method as above leads us to inequality (32) for the case \(-\pi<\theta<0\) with \(\theta\neq-\pi/2\) and the proof of Theorem 2.27 is now completed. □
Remark 2.28
Taking p to be appropriate analytic functions in Theorems 2.26 and 2.27, we can find conditions for univalence, starlikeness, convexity, and so on.
Theorem 2.29
Let \(0\leq{\alpha}<1\) and \(0< {\beta}\leq1\). If p is analytic in \(\mathbb{U}\) with \(p(0)=1\) and
where
and
then \(p\in\mathcal{P}({\alpha})\).
Proof
First, we note that \(p(z) \neq{\alpha}\) for \(0\leq{\alpha}<1\). Defining function w by (4), we see that w is analytic in \(\mathbb{U}\) with \(w(0)=0\). Suppose that there exists a point \(z_{0}\) in \(\mathbb{U}\) satisfying (5). Then we have (6). By using Lemma 2.1, we obtain
where k is a real number with \(k \geq1\). Putting \(p(z_{0})=\alpha +\mathrm {{i}}y\) with \(y\in\mathbb{R}\setminus\{0\}\), we obtain (10). Then we have
At first, we consider the case \(0<{\beta}<1\).
-
(i)
For the case \(y>0\), we have
$$ \begin{aligned} &\operatorname {Re}\biggl\{ \bigl(p(z_{0})-{\alpha} \bigr)^{{\beta}} \biggl( 1+\frac{{z_{0}p^{\prime}(z _{0})}}{{p(z_{0})-{\alpha}}} \biggr) \biggr\} \\ &\quad=\operatorname {Re}\biggl\{ \biggl( - \frac{k}{2(1-{\alpha})} \bigl( {(1-{\alpha})^{2}}y ^{{\beta}-1}+y^{{\beta}+1} \bigr) +\mathrm {{i}}y^{{\beta}} \biggr) \biggl( \sin\biggl( \frac{\pi}{2}{\beta}\biggr) - \mathrm {{i}}\cos\biggl( \frac{\pi }{2}{\beta}\biggr) \biggr) \biggr\} \\ &\quad= - \frac{k}{2(1-{\alpha})} \bigl( {(1-{\alpha})^{2}}y^{{\beta}-1}+y^{{\beta}+1} \bigr) \sin\biggl( \frac{\pi}{2}{\beta}\biggr) + y^{{\beta}} \cos\biggl( \frac{\pi}{2}{\beta}\biggr) \\ &\quad\leq\frac{1}{2(1-{\alpha})} \biggl( -y^{{\beta}+1} \sin\biggl( \frac{ \pi}{2}{\beta}\biggr) +2 (1-{\alpha})y^{{\beta}} \cos\biggl( \frac{\pi}{2}{\beta}\biggr) - {(1-{\alpha})^{2}}y^{{\beta}-1} \sin \biggl( \frac{ \pi}{2}{\beta}\biggr) \biggr) \\ &\quad=h(y,{\alpha}, {\beta}). \end{aligned} $$
Then, by a simple calculation, we obtain
which is a contradiction to our assumption.
-
(ii)
For the case \(y<0\), we have
$$ \begin{aligned} &\operatorname {Re}\biggl\{ \bigl(p(z_{0})-{\alpha} \bigr)^{{\beta}} \biggl( 1+\frac{{z_{0}p^{\prime}(z _{0})}}{{p(z_{0})-{\alpha}}} \biggr) \biggr\} \\ &\quad= \operatorname {Re}\biggl\{ \biggl( - \frac{k}{2(1-{\alpha})} \bigl( {(1-{\alpha})^{2}} \vert y \vert ^{{\beta}-1}+ \vert y \vert ^{{\beta}+1} \bigr) -\mathrm {{i}}\vert y \vert ^{{\beta}} \biggr) \biggl( \sin\biggl( \frac{\pi}{2} {\beta}\biggr) + \mathrm {{i}}\cos\biggl( \frac{ \pi}{2}{\beta}\biggr) \biggr) \biggr\} \\ &\quad=- \frac{k}{2(1-{\alpha})} \bigl( {(1-{\alpha})^{2}} \vert y \vert ^{{\beta}-1}+ \vert y \vert ^{{\beta}+1} \bigr) \sin\biggl( \frac{ \pi}{2}{\beta}\biggr)+ \vert y \vert ^{{\beta}} \cos\biggl( \frac{\pi}{2}{\beta}\biggr) \\ &\quad\leq h\bigl( \vert y \vert ,{\alpha}, {\beta}\bigr) \leq h\bigl({\delta}({\alpha}, {\beta}),{\alpha}, {\beta}\bigr). \end{aligned} $$
We also come up to the same contradiction to our assumption under \(y<0\) condition. Now, we consider the case \({\beta}=1\) and obtain
This contradicts our assumption. So, the proof is completed. □
Remark 2.30
Taking \({\alpha}=0\) and \({\beta}=1\) in Theorem 2.29, we obtain the same result of Corollary 2.22.
Taking \(p(z) = f(z)/z\) and \({\alpha}=0\) in Theorem 2.29, we have the following result.
Corollary 2.31
Let \(f\in\mathcal{A}\) and \(0< {\beta}\leq1\). If
where h and \({\delta}(0,{\beta})\) are given in Theorem 2.29, respectively, then
Example 2.32
Taking \({\beta}= 1/2\) in Corollary 2.31, we have \(h({\delta}(0,{\beta}),0,{\beta})=0\). Then
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The authors would like to express their gratitude to the referees for many valuable suggestions regarding a previous version of this paper.
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The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT & Future Planning) (No. NRF-2017R1C1B5076778). The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450).
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Kim, I.H., Sim, Y.J. & Cho, N.E. New criteria for Carathéodory functions. J Inequal Appl 2019, 13 (2019). https://doi.org/10.1186/s13660-019-1970-5
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DOI: https://doi.org/10.1186/s13660-019-1970-5