On a class of analytic functions closely related to starlike functions with respect to a boundary point

1.   Introduction

Consider $\mathbb{D} = \left\{z: |z| < 1\right\}$, an open unit disc in the complex plane $\mathbb{C}$, $\mathcal{H}$ as the collection of all analytic functions in $\mathbb{D}$. Let $\mathcal{A}$ be the subclass of $\mathcal{H}$ consisting of all functions of the form

and $\mathcal{S}$ be a subclass of $\mathcal{A}$ containing all univalent functions. A function $f\in\mathcal{A}$ is starlike if $f(\mathbb{D})$ is starlike with respect to origin and the class of all starlike functions $f\in\mathcal{A}$ is denoted by $\mathcal{ST}$. Similarly, a function $f\in\mathcal{A}$ is convex if $f(\mathbb{D})$ is convex with respect to all points of $f(\mathbb{D})$ and the class of all convex functions $f\in\mathcal{A}$ is denoted by $\mathcal{CV}$. An analytic function $p:\mathbb{D}\rightarrow \mathbb{C}$ of the form

satisfying $\Re \left(p(z)\right) > 0$ for all $z\in\mathbb{D}$ is known as a function with positive real part. The class of such functions, denoted by $\mathcal{P}$, is known as the class of Carathéodory functions. Note that $\Re \left(p(z)\right) > 0$ can be written as $|\arg\left(p(z)\right)| < \frac{\pi}{2}$. Connections between $\mathcal{ST}$ and $\mathcal{P}$ and $\mathcal{CV}$ and $\mathcal{P}$ are as follows: a function $f\in\mathcal{ST}$ if and only if $\frac{ z f^{\prime}(z)}{f(z)}\in\mathcal{P}, \, \, z\in\mathbb{D}$ and a function $f\in\mathcal{CV}$ if and only if $1+\frac{z f^{\prime \prime}(z)}{f^{\prime}(z)}\in \mathcal{P}, \, \, z\in\mathbb{D}$. Thus, the properties of $\mathcal{ST}$ and $\mathcal{CV}$ functions can be obtained from the properties of functions in the class $\mathcal{P}$. Note that the Möbius function

is analytic and univalent in the open unit disc $\mathbb{D}$ and it maps $\mathbb{D}$ onto the right half-plane and is in the class $\mathcal{P}$.

Even though many authors extensively explored the concept of starlikeness of a given order for a long time, Robertson ^[16] was the pioneer in introducing the concept of an analytic univalent functions mapping an open unit disc onto a starlike domain with respect to the boundary point. He constructed the subclass $\mathcal{G^*}$ of $\mathcal{H}$ of functions $g, g(0) = 1$ mapping $\mathbb{D}$ onto a starlike domain with respect to $g(1) = \lim_{r\rightarrow1^{-}}g(r) = 0$ and $\Re \left(e^{i\rho}g(z)\right) > 0$ for some real $\rho$ and all $z\in\mathbb{D}$. Assume also that the constant function $1$ belongs to the class $\mathcal{G^*}$. He conjectured that the class $\mathcal{G^*}$ coincides with the class $\mathcal{G}$,

such that

and proved that $\mathcal{G}\subset\mathcal{G^*}$. Later, this conjecture was confirmed by Lyzzaik ^[11] who proved $\mathcal{G^*}\subset\mathcal{G}$. Furthermore, if $g\in\mathcal{G}, g\neq1$ then $g$ is univalent close-to-convex in $\mathbb{D}$, as proved by Robertson ^[16]. It is worth mentioning that the analytic characterization (1.4) was known earlier to Styer ^[19].

In ^[3], a class $\mathcal{G}(M), M > 1$, consisting of all analytic and non-vanishing functions of the form (1.3), such that

which is a closely related function to the class $\mathcal{G}$ was introduced by Jakubowski ^[3]. Here,

is the Pick function. The class $\mathcal{G}(1)$ was also considered in ^[3], where

Todorov ^[20] linked the class $\mathcal{G}$ with a functional $f(z)/(1-z)$ for $z\in\mathbb{D}$ and obtained a structural formula and coefficient estimates. Silverman and Silvia ^[17] introduced an interesting class $\mathcal{G\left(\beta\right)}\subset\mathcal{G^*}, 0 < \beta\leq1$, consisting of all analytic function $g$ of the form (1.3) and satisfying

Clearly, $\mathcal{G}\left(\frac{1}{2}\right) = \mathcal{G}$. For $-1 < A\leq1$ and $-A < B\leq1$, Jakubowski and Włodarczyk ^[4] defined the class $\mathcal{G}(A, B)$, of all $g$ of the form (1.3), satisfying $\Re \left(2z\frac{g'(z)}{g(z)}+\frac{1+Az}{1-Bz}\right) > 0, \, \, \, z\in\mathbb{D}$ (see also the work of Sivasubramanian ^[18]). Related works on the class $\mathcal{G}$ were considered earlier by [1, 6–10, 14]. We remark at this point, that the function

is univalent in $\mathbb{D}$. In this article, we are interested in introducing and investigating a new class as follows.

Definition 1.1. Let $\mathcal{G}_c$ be the class consisting of all functions of the form (1.3) satisfying

where $0 < c \leq2$.

If $c = 1$, the class $\mathcal{G}_1 = \mathcal{G}$ was introduced and investigated by Robertson ^[16]. For this new class of functions, we obtain representation theorem, interesting coefficient estimates and also certain differential subordination implications involving this new class.

Example 1.1. The function

is in the class $\mathcal{G}_c$, where $0 < c\leq2$.

Proof. Taking logarithm on both sides and by a simple differentiation, one can easily get

Hence,

implies $g_1\in \mathcal{G}_c$.

Example 1.2. The function

is in the class $\mathcal{G}_c$, where $0 < c\leq2$.

Proof. Taking logarithm on both sides and by a simple differentiation, one can easily get

Therefore,

implies that $g_2\in \mathcal{G}_c$.

Similarly, we can show that

Example 1.3. The function

is in the class $\mathcal{G}_c$, where $0 < c\leq2$.

Example 1.4. The function

is in the class $\mathcal{G}_c$, where $0 < c\leq2$.

The above examples show that there are many functions present in the class $\mathcal{G}_c$ proving that the class $\mathcal{G}_c$ is non empty.

2.   Representation theorems

Theorem 2.1. Let $0 < c \leq 2$. Furthermore, let

A function $g$ is in $\mathcal{G}_c$ if and only if there exists a starlike function $s \in \mathcal{ST}$ such that

Proof. Suppose that $s\in \mathcal{ST}$ and $g$ is given by (2.2). Then, $g$ is analytic and $g(0) = 1$. Therefore, from (1.5), we have for some function $g$ satisfying (2.2), there exists a starlike function $s$ such that

Hence,

By a simple differentiation followed by simplification we get,

where

Therefore,

Hence, $g\in \mathcal{G}_c$.

On the other hand, suppose $g\in \mathcal{G}_c$ and

Then, $s(0) = 0, s^{\prime}(0) = 1$ and

The above expression is positive as $g\in \mathcal{G}_c$ which implies $s \in \mathcal{ST}$.

For the choices of $c = 1$ and $c = 2$ we get the following corollaries as listed below.

Corollary 2.1. ^[16] A function $g$ is in $\mathcal{G}_1$ if and only if there exists a function $s\in \mathcal{ST}$ such that $\left(g(z)\right)^2 = \left(\frac{s(z)}{z}\right)\left(1-z\right)^2$.

Corollary 2.2. A function $g$ is in $\mathcal{G}_2$ if and only if there exists a function $s\in \mathcal{ST}$ such that $g(z) = \left(\frac{s(z)}{z}\right)^{\frac{1}{2}} \ \exp \left(\frac{-2}{1-z}\right)$.

Theorem 2.2. (Herglotz representation theorem) Let $0 < c \leq 2$ and let $g$ be an analytic function in $\mathbb{D}$ such that $g(0) = 1$. Then, $g\in \mathcal{G}_c$ if and only if

where $\mu (t)$ is a probability measure on $[-\pi, \pi]$.

Proof. Let $0 < c \leq 2$. If $g\in \mathcal{G}_c$, we can write

By a simple integration and simplification we get,

Upon simplification of the above equation, one can obtain (2.3). The converse part can be proved by similar lines as in the necessary part and hence the details are omitted.

For $c = 1$ and $c = 2$ we get the following corollaries.

Corollary 2.3. ^[16] Let $g$ be an analytic function in $\mathbb{D}$ such that $g(0) = 1$. Then, $g\in \mathcal{G}_1$ if and only if

Corollary 2.4. Let $g$ be an analytic function in $\mathbb{D}$ such that $g(0) = 1$. Then, $g\in \mathcal{G}_2$ if and only if

Theorem 2.3. Let $0 < c \leq 2$. A function $g \in \mathcal{G}_c$ if and only if there exists a function $p\in \mathcal{P}$ such that

Proof. Let $g \in \mathcal{G}_c$. Then, by the definition of $\mathcal{G}_c$, for some function $p \in \mathcal{P}$,

Upon integration and simplification, we get

which proves the necessary part of the theorem. Conversely, assume $p\in \mathcal{P}$ and $p(0) = 1$ and let $g$ be as in (2.4). Then, $g$ is analytic in $\mathbb{D}$ and by applying simple calculations we can easily prove that $g\in\mathcal{G}_c$.

For $c = 1$ and $c = 2$, we get the following corollaries.

Corollary 2.5. Let $g \in \mathcal{G}_1$ if and only if there exists a function $p\in \mathcal{P}$ such that

Corollary 2.6. Let $g \in \mathcal{G}_2$ if and only if there exists a function $p\in \mathcal{P}$ such that

Theorem 2.4. Let $0 < c \leq 2$. A function $g\in \mathcal{G}_c$ if and only if there exists a function $p\in \mathcal{H}$ such that $p\prec \frac{1+z}{1-z}$ and for $z\in\mathbb{D}$,

Proof. The proof follows from Theorem 2.3.

3.   Coefficient estimates for the class $\mathcal{G}_c$

Theorem 3.1. Let $0 < c \leq 2$ and $z\in \mathbb{D}$. If $g \in \mathcal{G}_c$, we have the following sharp inequalities.

and

Further, for $\alpha \in \mathbb{R}$, let

Then,

Proof. Let

On expanding the right hand side of the above function $p$, we get

By making use of the known inequality $|p_i| \leq 2$ for all $p \in \mathcal{P}$, we can get the sharp inequalities given in (3.1)–(3.3). From (1.1) and (3.6) and from the known fact that

we can obtain (3.5)

For $c = 1$ and $c = 2,$ we have the following corollaries as stated below.

Corollary 3.1. ^[2] Let $z\in \mathbb{D}$. If $g \in \mathcal{G}_1$, we have the following inequalities.

Further,

All of these inequalities are sharp.

Corollary 3.2. Let $z\in \mathbb{D}$. If $g \in \mathcal{G}_2$, the following inequalities hold.

Also,

All of these inequalities are sharp.

Theorem 3.2. Let $0 < c \leq 2$ and let the function $g(z)$ be of the form (1.3) belong to the class $\mathcal{G}_c$. Then, for $n = 2, 3, \cdots$, the following estimates

hold.

Proof. Let the function $g$ of the form (1.3) belong to the class $\mathcal{G}_c$. Then, there exists a function $p\in \mathcal{P}$ such that,

Since $p\in \mathcal{P}$, there exists a function $\omega$ of the form

where $\omega$ is analytic in $\mathbb{D}, \omega\left(0\right) = 0, \left|\omega \left(z\right)\right|\leq 1$ for $z\in\mathbb{D}$. Furthermore,

From (3.7) and (3.8), we have

Let

Considering the expansion of the function of $g$ as power series, we get

$(2+\sum_{n = 1}^{\infty}\left(d_n-cd_{n-1}+\frac{c(c-1)}{2!}d_{n-2}+\cdots+(-1)^{n-1} \, \frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1+\\ (-1)^{n} \, \frac{c(c-1)\cdots(c-n+1)}{n!}\right)z^n\\+2\sum_{n = 1}^{\infty}\left(nd_n-c(n-1)d_{n-1}+\frac{c(c-1)}{2!}(n-2)d_{n-2}+\cdots+(-1)^{n-1} \, \frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1\right)z^n\\+ \sum_{n = 1}^{\infty}\left(d_n+cd_{n-1}+\frac{c(c-1)}{2!}d_{n-2}+\cdots+\frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1+\frac{c(c-1)\cdots(c-n+1)}{n!}\right)z^n)\left(\sum_{n = 1}^{\infty}\omega_n z^n\right)\\ = 2\sum_{n = 1}^{\infty}\left(nd_n-c(n-1)d_{n-1}+\frac{c(c-1)}{2!}(n-2)d_{n-2}+\cdots+(-1)^{n-1} \, \frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1 \right)z^n\\+ \sum_{n = 1}^{\infty}\left(d_n+cd_{n-1}+\frac{c(c-1)}{2!}d_{n-2}+\cdots+\frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1 +\frac{c(c-1)\cdots(c-n+1)}{n!}\right)z^n\\- \sum_{n = 1}^{\infty}\left(d_n-cd_{n-1}+\frac{c(c-1)}{2!}d_{n-2}+\cdots+(-1)^{n-1} \, \frac{c(c-1)\cdots(c-n+2)}{(n-1)!}d_1+(-1)^{n} \, \frac{c(c-1)\cdots(c-n+1)}{n!}\right)z^n$.

For $z \in \mathbb{D}$, this can be again simplified to bring into the form,

For $\ n = 2, 3, \cdots$, let

and

From (3.9)–(3.11), we have

Equating the coefficient of $z$, we have

Since $|\omega_1|\leq1$, we obtain

and for $n = 2, 3, \cdots$,

From the Eqs (3.9)–(3.12), we have

where $E_k$ are the appropriate coefficients. Since $|\omega(z)| < 1$ for $z\in\mathbb{D}$,

By simplifying this, we have

Since $|s_k(c)|^2\geq0$ for $k = 1, 2, \cdots, n-1,$ we obtain

This essentially completes the proof of Theorem 3.2.

Let us consider the class $\mathcal{B}$ defined by $\mathcal{B} = \left\{\omega \in\mathcal{H}: \left|\omega (z)\right|\leq 1, z\in \mathbb{D}\right\}$ and $\mathcal{B}_0$ be the subclass of $\mathcal{B}$ consisting of functions $\omega$ such that $\omega(0) = 0.$ The elements of $\mathcal{B}_0$ are also known as Schwarz functions.

Lemma 3.1. ^[5] If $\omega \in \mathcal{B}_0$ is of the form $\omega(z) = \sum\limits_{n = 1}^{\infty}\omega_n z^n, \ z\in\mathbb{D},$ then for $\nu\in\mathbb{C}$,

Lemma 3.2. If $\omega\in\mathcal{B}_{0}$ is of the form $\omega(z) = \sum\limits_{n = 1}^{\infty}\omega_n z^n, \ z\in\mathbb{D},$ then for any real numbers $q_1$ and $q_2$, the following sharp estimates holds:

where

and the sets $D_k, k = 1, 2, \cdots$are defined in ^[15].

Now we obtain a few upper bounds for early coefficients and for the Fekete-Szegö functional in the class $\mathcal{G}_c$.

Theorem 3.3. Let $g\in\mathcal{G}_c, 0 < c\leq2$. Then,

and

Furthermore, for $\delta\in\mathbb{R}$

Proof. From the class $\mathcal{G}_c$, there exists $\omega\in\mathcal{B}_0$ of the form $\omega(z) = \sum\limits_{n = 1}^\infty\omega_n z^n, \ z\in\mathbb{D}$, such that

Since

and

By comparing the corresponding coefficients from (3.21)–(3.23), we have the following:

and

Since $|\omega_1|\leq1$, from (3.24),

and hence

From (3.25) and by Lemma 3.1, we have

and

which in turn gives

and hence

From (3.26) and by Lemma 3.2, we get

Hence, we obtain

Substituting the formulas for $d_1$ and $d_2$, we obtain

Therefore,

Furthermore, we get

The proof of the theorem is completed by virtue of Lemma 3.1.

It can be remarked here that if $\nu$ is a real number, Lemma 3.1 can be improved in the following way and can be found in ^[2].

Lemma 3.3. If $\omega \in \mathcal{B}_0$ is of the form $\omega(z) = \sum\limits_{n = 1}^{\infty}\omega_n z^n, \ z\in\mathbb{D},$ then

For $\nu < -1$ or $\nu > 1$, equality holds if and only if $\omega\left(z\right) = z$ or one of its rotations. For $-1 < \nu < 1$, equality holds if and only if $\omega(z) = z^2, \ z\in\mathbb{D}$ or one of its rotations. For $\nu = -1$ equality holds if and only if $\omega(z) = \frac{z(\lambda+z)}{(1+\lambda z)}, z\in\mathbb{D}$ or one of its rotations, while for $\nu = 1$ equality holds if and only if $w(z) = \frac{-z(\lambda+z)}{(1+\lambda z, )}, 0\leq\lambda\leq1, z\in\mathbb{D}$ or one of its rotations.

We can improve the results obtained in (3.20), in view of Lemma 3.3 as follows: For $\delta\in\mathbb{R}$, we get,

Theorem 3.4. Let $0 < r < 1$. If $g\in\mathcal{G}_c$ then for $|z| = r < 1$,

Proof. Let us define

Note that the function $g$ is non-vanishing in $\mathbb{D}$. Therefore, $f$ is analytic in $\mathbb{D}$ and a simple computation shows that

From (3.31), we can see that $g\in \mathcal{G}_c$ if and only if $f\prec\frac{1+z}{1-z}, z\in\mathbb{D}$. By applying the result of Ma and Minda ^[12], we infer that

Hence,

which gives (3.29).

For $c = 1$ and $c = 2$ we have the following corollaries.

Corollary 3.3. For $0 < r < 1$, if $g\in\mathcal{G}_1$ then we have for $|z| = r < 1$

Corollary 3.4. For $0 < r < 1$, if $g\in\mathcal{G}_2$ then we have for $|z| = r < 1$,

Theorem 3.5. Let $g\in \mathcal{G}_c$. Then,

and

Proof. Since $g\in \mathcal{G}_c$, there is a Schwarz function $\omega$ satisfying that

If $\omega(z) = z$, we have

Equating the coefficients using (3.22),

Since $0 < c\leq2$, (3.36) holds.

Also, since $0 < c\leq2$,

implies that (3.37) holds.

4.   Differential subordination involving the class $\mathcal{G}_c$

In this section we are connecting the class $\mathcal{G}_c$, $0 < c\leq2$ by taking the equivalent condition of $\mathcal{G}_c$, $0 < c\leq2$ associated with $L_0(z) = \frac{1+z}{1-z}$. Note that the definition $\mathcal{G}_c, 0 < c\leq2$ can be rewritten in the equivalent form as $g \in \mathcal{G}_c$ if

To prove the differential subordination results, we need the following lemma which is stated below.

Lemma 4.1. ^[13] Let $\tau$ be univalent in $\mathbb{D}$, $\psi$ and $\phi$ be analytic in a domain $D$ containing $\tau(\mathbb{D})$ with $\phi(\omega)\neq0$ when $\omega\in \tau(\mathbb{D})$. Let $T(z) = z \tau' \phi(\tau(z))$ and $\kappa(z) = \psi(\tau(z))+T(z)$ for $z\in \mathbb{D}$ and satisfy either $T$ is starlike univalent in $\mathbb{D}$ or $\kappa$ is convex univalent in $\mathbb{D}$. Also, assume that $\Re \left\{\frac{z\kappa'(z)}{T(z)}\right\} > 0, z\in \mathbb{D}.$\\ If $p\in \mathcal{H}$ with $p(0) = \tau(0), p(\mathbb{D})\subset D$, and

then $p\prec\tau$ and $\tau$ is the best dominant.

Theorem 4.1. Let $g$ be an analytic function with $g(0) = 1$ and let $0 < c\leq2$. If $g$ satisfies

then

Proof. Let $\psi(\omega) = 1, \omega\in\mathbb{C}$ and $\phi(\omega): = \frac{1}{\omega}, \omega\in \mathbb{C}\setminus \left\{0\right\}$. Note that $L_0(\mathbb{D}): = \mathbb{C}\setminus\left\{0\right\}$. Thus,

is analytic and also well defined. Also, we have

This implies that $T$ is a starlike univalent function. From this, for a function $\kappa(z): = \psi(L_0(z))+T(z) = 1+T(z), z\in \mathbb{D}$, we get

From Lemma 4.1,

which implies that $p\prec L_0$. Let us take the analytic function $g$ with $g(0) = 1$ and $g(z)\neq0$ for $z\in\mathbb{D}$ satisfying (4.2). For a function as in (4.3), we can notice that $p(0) = L_0(0) = 1, p(z)\neq0$ for $z\in\mathbb{D}$ and $p$ is analytic. The proof of Theorem 4.1 is completed by observing that

Theorem 4.2. Let $g(z)$ be an analytic function with $g(0) = 1$ and let $0 < c\leq2$. If

then

Proof. Let $\psi(\omega) = \omega, \omega\in\mathbb{C}$ and $\phi(\omega): = \frac{1}{\omega}, \omega\in \mathbb{C}\setminus \left\{0\right\}$. Note that $L_0(\mathbb{D}): = \mathbb{C}\setminus \left\{0\right\}$ and $\psi$ and $\phi$ are analytic in $D$. Thus, $T$ defined by (4.4) is analytic and univalent and satisfies (4.5). Hence, $\kappa(z) = \psi(L_0(z))+T(z) = L_0(z)+T(z), \ z\in \mathbb{D}$. By using (4.5) we get,

Note that, $p$ is also analytic with $p(0) = L_0(0) = 1$ and $p(z)\neq0$ for $z\in\mathbb{D}$. From Lemma 4.1, we have

This essentially gives us that $p\prec L_0$. Let us take the analytic function $g$ with $g(0) = 1$ and $g(z)\neq0$ for $z\in\mathbb{D}$ satisfying (4.6). For a function defined as in (4.7), we can observe that

Therefore, $p(z)\neq0$ and analytic for all $z\in \mathbb{D}$. The proof of the theorem is then completed by noting that

Use of AI tools declaration

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

Acknowledgments

The authors are thankful to the referee(s) for many insightful suggestions and comments in the original manuscript. The second 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 competing interests.