First-order differential subordinations associated with Carathéodory functions

1.   Introduction

Let $\mathcal{P}(\alpha)$ be the class of analytic functions $p$ of the form $p(z) = 1+\sum_{n = 1}^{\infty}p_nz^n$ in the open unit disk ${{\mathbb U}} = \{z\in {\mathbb C} : |z| < 1 \}$, with ${\rm{Re}} p(z) > \alpha$ for $z\in\mathbb{U}$. The class $\mathcal{P}\equiv \mathcal{P}(0)$ is known as the Carathéodory class or the class of functions with positive real part ^[2,3], pioneered by Carathéodory. The theory of Carathéodory functions plays a very important role in the geometric function theory. For recent developments, the readers may refer to the works of Kim and Cho ^[5], Kwon and Sim ^[6], Nunokawa et al. ^[16], Sim et al. ^[18] and Wang ^[22].

Let $\cal A$ denote the class of all functions $f$ analytic in ${{\mathbb U}}$ with the usual normalization $f(0) = f^{\prime}(0) - 1 = 0$. If $f$ and $g$ are analytic in ${{\mathbb U}}$, we say that $f$ is subordinate to $g$, written $f \prec g$ or $f(z) \prec g(z)$, if there exists a Schwarz function $w(z)$ in ${{\mathbb U}}$ such that $f(z) = g(w(z))$.

A function $f \in {\cal A}$ is said to be strongly starlike of order ${{\eta}} \ (0 < {{\eta}} \leq 1)$ if, and only if,

We note that the conditions (1.1) can be written by

We denote by ${\cal S}[{{\eta}}]$ the subclass of $\cal A$ consisting of all strongly starlike functions of order ${{\eta}} \ (0 < {{\eta}} \leq 1)$. The class ${\cal S}[{{\eta}}]$ was introduced and studied by Brannan and Kirwan ^[1] and Stankiewicz ^[20,21]. We also note that ${\cal S}[1]\equiv {\cal S}^*$ is the well-known class of all normalized starlike functions in ${{\mathbb U}}$. The class ${\cal S}[{{\eta}}]$ and the related classes have been extensively studied by Mocanu ^[14] and Nunokawa ^[15]. It is worth noticing that $f$ belongs to ${\cal S}[{{\eta}}]$ if it satisfies

where

Given $\alpha \in [0, 1)$, let ${\cal S}^*(\alpha)$ be the subclass of ${\cal A}$, which consists of all starlike functions of order $\alpha$, namely, $f \in {\cal A}$ belongs to ${\cal S}^*(\alpha)$ if, and only if, it satisfies

The class ${\cal S}^*(\alpha)$ was introduced by Robertson ^[17]. Clearly, it holds that ${\cal S}^*(0) \equiv {\cal S}[1]\equiv {\cal S}^*$. A typical sufficient condition for starlike functions of order $\alpha$ is given by Wilken and Feng ^[23], which states that if $f \in {\cal A}$, then

implies $f \in {\cal S}^*(\alpha)$, where

Given ${{\eta}} \in (0, 1]$, let ${\cal T}[{{\eta}}]$ be the class of $f \in {\cal A}$ such that

The class ${\cal T} \equiv {\cal T}[1]$ plays an important role in the theory of univalent functions, although all elements in ${\cal T}$ are functions that are not necessarily univalent. In ^[7], several sufficient conditions for functions in ${\cal T}[{{\eta}}]$ were introduced.

If $\psi$ is analytic in a domain ${\mathbb D} \subset {\mathbb C}^2$, $h$ is univalent in ${{\mathbb U}}$ and $p$ is analytic in ${{\mathbb U}}$ with $(p(z), zp^{\prime}(z)) \in {\mathbb D}$ for $z \in {{\mathbb U}}$, then $p$ is said to satisfy the first-order differential subordination if

The univalent function $q$ is said to be a dominant of the differential subordination (1.2) if $p \prec q$ for all $p$ satisfying (1.2). If $\tilde q$ is a dominant of (1.2) and $\tilde q \prec q$ for all dominants of (1.2), then $\tilde q$ is said to be the best dominant of the differential subordination (1.2). The general theory of the first-order differential subordinations, with many interesting applications, especially in the theory of univalent functions, was developed by Miller and Mocanu ^[10] (also see ^{[4,8,9,11,12,13]}).

In this paper, by applying the result obtained by Miller and Mocanu ^[10], we will investigate conditions to be in the class of Carathéodory functions. We will also find new sufficient conditions for $f \in {\cal A}$ to belong to the classes ${\cal S}[{{\eta}}]$, ${\cal S}^*(\alpha)$, and ${\cal T}[{{\eta}}]$ as some applications of the main results presented here. A differential subordination of the Briot-Bouquet type ^[12] (also see ^[13]HY__HY, Section 3]) will be considered for conditions for $f \in {\cal S}^*(\alpha)$ and $f \in {\cal T}[1]$, and an integral operator related to the differential subordination of this type will be discussed as our additional results. Moreover, more conditions for $f \in {\cal S}[{{\eta}}]$ and $f \in {\cal T}[{{\eta}}]$ will be introduced by using a nonlinear first-order differential subordination.

2.   Main results

In proving our results, we shall need the following lemma due to Miller and Mocanu ^[10].

Lemma 1. Let $q$ be univalent in ${{\mathbb U}}$ and let $\theta$ and $\varphi$ be analytic in a domain $\mathbb D$ containing $q({{\mathbb U}})$ with $q(\omega) \not = 0$ when $\omega \in q({{\mathbb U}})$. Set $Q(z) = zq^{\prime}(z) \varphi(q(z)), h(z) = \theta(q(z)) + Q(z)$ and suppose that

(ⅰ) $Q$ {is starlike in} ${{\mathbb U}}$,

(ⅱ) $\mathrm{Re}\; \left\{ \frac{zh^{\prime}(z)}{Q(z)} \right\} = \mathrm{Re}\; \left\{ \frac{\theta^{\prime}(q(z))}{\varphi(q(z))} + \frac{zQ^{\prime}(z)}{Q(z)} \right\} > 0 \quad (z \in {{\mathbb U}}).$

If $p$ is analytic in ${{\mathbb U}}$ with $p(0) = q(0), \ p({{\mathbb U}}) \subset {\mathbb D}$, and

then $p \prec q$ and $q$ is the best dominant of (2.1).

With the help of Lemma 1, we now derive the following Theorem 1.

Theorem 1. Let $p$ be analytic in ${{\mathbb U}}$ with $p(0) = 1$ and ${{\beta}} > 0, \ {{\beta}}+{{\gamma}} > 0$. If

for all $k \ (|k| \geq {\sqrt{2({{\beta}} + {{\gamma}}) +1}/ {{\beta}}})$, then

Proof. First, we note that $p(z) \not = -({{\gamma}}/ {{\beta}})$ for $z\in {{\mathbb U}}$ under the condition (2.2). In fact, if ${{\beta}} p(z)+{{\gamma}}$ has a zero $z_0 \in {{\mathbb U}}$ of order $n \ (n\geq 1)$ at a point $z_0\in {{\mathbb U}}\backslash\{0\}$, then we may write

where $p$ is analytic in ${{\mathbb U}}$ with $q(z_{0}) \not = 0$, then it follows that

Therefore,

Letting $z$ approach $z_0$ in the direction of $\arg z_0$, the righthand side of (2.3) takes infinite pure imaginary value. This contradicts the assumption (2.2).

Let $q(z) = {\left({1+(1+{2{{\gamma}}/{{\beta}}})z}\right) / (1-z)}, \ {{\theta}}({{\omega}}) = {{\omega}}$, and ${{\varphi}}({{\omega}}) = 1/({{\beta}} {{\omega}} + {{\gamma}})$ in Lemma 1, then ${{\theta}}$ and ${{\varphi}}$ are analytic in $q({{\mathbb U}})$ and ${{\varphi}}({{\omega}}) \not = 0$ for ${{\varphi}} \in q({{\mathbb U}})$. Setting

and

the conditions (ⅰ) and (ⅱ) of Lemma 1 can be verified. Therefore, Lemma 1 gives that if

with

then

Noting that

we obtain

and

Meanwhile, since the imaginary part of $h(e^{i{{\theta}}})$ is an odd function, we consider only the case $0 < {{\theta}} < \pi$. Putting $\tan {({{\theta}}/2)} = t \ (0 < {{\theta}} < \pi)$, we have

Here, the function $g(t)$ has a minimum value at $t_{0} = \sqrt{2({{\beta}} + {{\gamma}}) + 1}$. Hence we have

Applying Lemma 1 and the assumption (2.2), we conclude that

This completes the proof of Theorem 1. □

Taking $p(z) = zf^{\prime}(z)/f(z), \ {{\beta}} = 1$, and ${{\gamma}} = (1/{{\alpha}})-1 \ (0 < {{\alpha}} \leq 1)$ in Theorem 1, we have the following result.

Corollary 1. Let $f \in \cal A$ and $0 < {{\alpha}} \leq 1$. If

for all $k \ (|k| \geq \sqrt{(2+{{\alpha}})/{{\alpha}}}/{{\beta}})$, then

Proof. Putting

we have

and

Hence,

Therefore, applying Theorem 1, we have Corollary 1. □

Corollary 2. Let $f \in \cal A$ and let

If

for all $k \ (|k| \geq \sqrt{(2+{{\alpha}})/{{\alpha}}}/{{\beta}})$, then

Proof. Differentiating $F$ with respect to $z$ and multiplying by $z$, we have

Therefore, the result follows from Corollary 1. □

Letting ${{\beta}} = {1/{{\alpha}}} \ ({{\alpha}} > 0), \ {{\gamma}} = 0$, and $p(z) = {zf^{\prime}(z)/f(z)}$ in Theorem 1, we have the following result.

Corollary 3. Let $f\in {\cal A}$ and ${{\alpha}} > 0$. If

for all $k\ (|k| \geq \sqrt{{{\alpha}}(2+{{\alpha}})})$, then $f$ is starlike in ${{\mathbb U}}$.

Taking ${{\beta}} = 1, \ {{\gamma}} = 0$, and $p(z) = zf^{\prime}(z)/f(z)$ in Theorem 1, we have the following result.

Corollary 4. Let $f\in {\cal A}$. If

for all $k \ (|k| \geq \sqrt{3})$, then $f$ is a starlike in ${{\mathbb U}}$.

Example 1. Consider a function $\tilde{f} : \mathbb{U} \rightarrow \mathbb{C}$ defined by

Then we have

and

Therefore, by Corollary 4, $\tilde{f}$ is starlike in $\mathbb{U}$ (see also the left side of Figure 1). In fact, we can check that $\mathrm{Re}\; \{ z\tilde{f}'(z) / \tilde{f}(z) \} > 0$ holds for all $z\in\mathbb{U}$, as shown in the right side of Figure 1.

Letting ${{\beta}} = 1, \ {{\gamma}} = 0$ and $p(z) = {f(z)/z}$ in Theorem 1, we have the following result.

Corollary 5. Let $f\in {\cal A}$. If

for all $k$ with $|k| \geq \sqrt{3}$, then

Further, we derive the following corollary.

Corollary 6. Let $f\in {\cal A}$ and let

If

for all $k \ (|k| \geq \sqrt{2({{\beta}}+ {{\gamma}}) +1}/ {{\beta}})$, then

Proof. From the definition of $F$, we have

Let

Taking logarithmic derivatives in (2.4) and multiplying by $z$, we obtain, after some simple calculations,

Therefore, applying Theorem 1, we have the result. □

Next, we prove the following theorem.

Theorem 2. Let $p$ be nonzero analytic in ${{\mathbb U}}$ with $p(0) = 1$ and $\ 0 < {{\eta}} < 1$. If

where

and

then

Proof. We choose $q(z) = {\Big((1+z)/(1-z)\Big)}^{{{\eta}}} \ (0 < {{\eta}} < 1), \ \theta (\omega) = 1 - {1/\omega}$, and $\varphi (\omega) = {1/\omega^{2}}$ in Lemma 1, then we see that $\theta$ and $\varphi$ are analytic in $q({{\mathbb U}})$ and $\varphi (\omega) \not = 0$ for $\omega \in q({{\mathbb U}})$. Further,

is starlike, and for the function

we have

Note that $h(0) = 0$ and

where we take " + " for $0 < \theta < \pi$, and " - " for $-\pi < \theta < 0$. Since the imaginary part of $h(e^{i \theta})$ is an odd function of $\theta$, we consider only the case $0 < \theta < \pi$. If we put $\tan {(\theta /2)} = t \ (t > 0)$, then we have

It is easy to see that the function $g(t)$ has the minimum value at the point

Therefore, we conclude that

and so, by assumption (2.5),

Hence, from Lemma 1, we have $p(z) \prec q(z) \ (z \in {{\mathbb U}})$, and this completes the proof of Theorem 2. □

From Theorem 2, we have the following result.

Corollary 7. Let $f \in {\cal A}$ with ${f(z) f^{\prime}(z)/z} \not = 0$ for $z \in {{\mathbb U}}$ and $0 < {{\eta}} < 1$. If

where $C({{\eta}})$ is given by (2.6), then

Proof. Setting

in Theorem 2, we see that $p$ is regular in ${{\mathbb U}}, \ p(0) = 1$, and $p(z) \not = 0$ in ${{\mathbb U}}$. It can be derived that

Thus, from Theorem 2, we immediately have the result. □

Example 2. Letting ${{\eta}} = {1/2}$ in Corollary 7, we have $C(1/2) \doteqdot 0.72674$. Therefore, if

then

Taking $p(z) = f(z)/z$ in Theorem 2, we have the following corollary.

Corollary 8. Let $f \in {\cal A}$ with ${f(z)/z} \not = 0$ for $z \in {{\mathbb U}}$ and $0 < {{\eta}} < 1$. If

where $C({{\eta}})$ is given by (2.6), then

Finally, by using a similar method of the proofs of Theorems 1 and 2, we have Theorem 3 below.

Theorem 3. Let $\alpha$, $\beta$, and $\eta$ be real numbers satisfying $\alpha > 0$, $0 < \eta \leq 1$, and

where

Let $p$ be analytic in ${{\mathbb U}}$ with $p(0) = 1$. If

then

Proof. We note that the inequality (2.9) is well-defined by (2.7). Applying the same method of the proof in Theorem 1, we can see that $p(z) \not = 0$ for $z\in {{\mathbb U}}$. Let $q(z) = {\Big((1+z)/(1-z) \Big)}^{{{\eta}}} \ (0 < {{\eta}} \leq 1), \ \theta (\omega) = \omega - {{\beta}}$, and $\varphi (\omega) = {{{\alpha}}/\omega}$ in Lemma 1, then

and

Also, the other conditions (ⅰ) and (ⅱ) of Lemma 1 can be checked to be satisfied. Note that

and

Setting $t = \cot \ {(\theta/2)} \ (0 < \theta < \pi)$ without loss of generality, we obtain

We first consider the case $\beta \cos (\pi\eta/2) > \alpha \eta \sin (\pi\eta/2)$, then the function $g(t)$ has the minimum value at

so that

Hence we see that

Therefore, by the assumption (2.9), we have

Next, we consider the case $\beta \cos (\pi\eta/2) \leq \alpha \eta \sin (\pi\eta/2)$, then the function $g$ is increasing on $(0, \infty)$ and it follows that

Hence, we get

Therefore, by the assumption (2.9), we have (2.10) again. Finally, with the aid of Lemma 1, we obtain $p(z) \prec q(z) \ (z \in {{\mathbb U}})$, that is, $|\arg \ p(z)| < {\pi \over 2}{{\eta}}$. □

Taking ${{\beta}} = {{\alpha}}$ in Theorem 3, we have the following result.

Corollary 9. Let $\alpha$ and $\eta$ be real numbers such that $\alpha > 0$, $0 < \eta \leq 1$, and

Let $x^* = 0.638\ldots$ be the unique root of the equation $x = \cot (\pi x/2)$. If $f \in {\cal A}$ satisfies

where

then

Example 3. Choosing ${{\alpha}} = 1$ and ${{\eta}} = {1 / 2}$ in Corollary 9, we have $C(1, \ {1/ 2}) = {{3 \sqrt 2} / {4}}$. Therefore, we obtain that if

then

Making $p(z) = {f(z)/ z}$ in Theorem 3, we have the following result.

Corollary 10. Let $\alpha$, $\beta$, and $\eta$ be real numbers satisfying (2.7). If $f\in {\cal A}$ satisfies

where $C({{\alpha}}, \ {{\beta}}, \ {{\eta}})$ is given by (2.8), then

We remark that, for the case $\eta = 1$ in Theorem 3, we have $C({{\alpha}}, \ {{\beta}}, \ 1) = \sqrt{\alpha^2+\beta^2}$. We end this paper with showing that this quantity can be improved as follows:

Corollary 11. Let $\alpha$ and $\beta$ be real numbers such that $\alpha > 0$ and $\sqrt{\alpha(\alpha+2)+\beta^2} > |1-\beta|$. Let $p$ be analytic in ${{\mathbb U}}$ with $p(0) = 1$. If

then $\mathrm{Re}\; p(z) > 0$ for all $z\in{{\mathbb U}}$.

Proof. By defining the same functions $q$, $\theta$, $\varphi$, $Q$, and $h$ with $\eta = 1$, as in the proof of Theorem 3, we will reach the following equality:

where $t = \cot(\theta/2)$ with $0 < \theta < \pi$. Furthermore, since $t > 0$, we get

Hence, combining (2.11) and (2.12) leads us to get

Thus, it follows from the same proof of Theorem 3 that $| \arg p(z) | < \pi/2$ ($z\in{{\mathbb U}}$), or $\mathrm{Re}\; p(z) > 0$ ($z\in{{\mathbb U}}$). □

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

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. 2019R1I1A3A01050861).

Conflict of interest

The authors declare that they have no conflicts of interest.