On a subclass related to Bazilevič functions

1.   Introduction and definitions

Let $\mathfrak{A}$ denote the family of all functions $f$ which are analytic in the open unit disc $\mathcal{U} = \left \{ z:\left \vert z\right \vert < 1\right \}$ and satisfying the normalization

while by $\mathcal{S}$ we mean the class of all functions in $\mathfrak{A}$ which are univalent in $\mathcal{U}.$ Also let $\mathcal{S}^{\ast }$ and $\mathcal{C}$ denote the familiar classes of starlike and convex functions, respectively. If $f$ and $g$ are analytic functions in $\mathcal{U },$ then we say that $f$ is subordinate to $g,$ denoted by $f\prec g,$ if there exists an analytic Schwarz function $w$ in $\mathcal{U}$ with $w\left(0\right) = 0$ and $\left \vert w(z)\right \vert < 1$ such that $f(z) = g\left(w(z)\right).$ Moreover if the function $g$ is univalent in $\mathcal{U}$, then

For arbitrary fixed numbers $A,$ $B$ and $b$ such that $A$, $B$ are real with $-1\leq B < A\leq 1$ and $b\in \mathbb{C} \backslash \{0\}$, let $\mathcal{P}[b, A, B]$ denote the family of functions

analytic in $\mathcal{U}$ such that

Then, $p\in \mathcal{P}[b, A, B]$ can be written in terms of the Schwarz function $w$ by

By taking $b = 1-\sigma$ with $0\leq \sigma < 1,$ the class $\mathcal{P} [b, A, B]$ coincides with $\mathcal{P}[\sigma, A, B],$ defined by Polatoğ lu ^[17,18] (see also ^[2]) and if we take $b = 1,$ then $\mathcal{ P}[b, A, B]$ reduces to the familiar class $\mathcal{P}[A, B]$ defined by Janowski ^[10]. Also by taking $A = 1,$ $B = -1$ and $b = 1$ in $\mathcal{P} [b, A, B]$, we get the most valuable and familiar set $\mathcal{P}$ of functions having positive real part. Let $\mathcal{S}^{\ast }[A, B, b]$ denote the class of univalent functions $g$ of the form

in $\mathcal{U}$ such that

Then $S^* [A, B] : = S^* [A, B, 1]$ and the subclass $\mathcal{S}^{\ast}[1,-1,1]$ coincides with the usual class of starlike functions.

The set of Bazilevič functions in $\mathcal{U}$ was first introduced by Bazilevič ^[7] in 1955. He defined the Bazilevič function by the relation

where $p\in \mathcal{P},$ $g\in \mathcal{S}^{\ast },$ $\beta$ is real and $\alpha > 0$. In 1979, Campbell and Pearce ^[8] generalized the Bazilevi č functions by means of the differential equation

where $\alpha +i\beta \in \mathbb{C} -\left \{ \text{negative integers}\right \}.$ They associate each generalized Bazilevič functions with the quadruple $(\alpha, \beta, g, p).$

Now we define the following subclass.

Definition 1.1. Let $g$ be in the class $\mathcal{S}^{\ast }[A, B]$ and let $p\in \mathcal{P} [b, A, B]$. Then a function $f$ of the form $\left(1.1 \right)$ is said to belong to the class of generalized Bazilevič function associated with the quadruple $(\alpha, \beta, g, p)$ if $f$ satisfies the differential equation

where $\alpha + i\beta \in \mathbb{C} - \{{\text {negative integers}} \}$.

The above differential equation can equivalently be written as

or

Since $p\in \mathcal{P}[b, A, B],$ it follows that

where $g\in \mathcal{S}^{\ast }[A, B].$

Several research papers have appeared recently on classes related to the Janowski functions, Bazilevič functions and their generalizations, see ^{[3,4,5,13,16,21,22]}.

2.   Lemmas

The following are some results that would be useful in proving the main results.

Lemma 2.1. Let $p\in \mathcal{P}[b, A, B]$ with $b\neq 0,$ $-1\leq B < A\leq 1,$ and has the form $\left(1.2\right).$ Then for $z = re^{i\theta },$

Proof. The proof of this lemma is straightforward but we include it for the sake of completeness. Since $p\in \mathcal{P}[b, A, B]$, we have

Let $\tilde{p}(z) = 1+\sum_{n = 1}^{\infty }c_{n}\, z^{n}$. Then

Comparing the coefficients of $z^{n}$, we have

Since $\left \vert c_{n}\right \vert \leq A-B$ ^[20], it follows that $\left \vert p_{n}\right \vert \leq \left \vert b\right \vert (A-B)$ and so

Thus the proof is complete.

Lemma 2.2. ^[1] Let $\Omega$ be the family of analytic functions $\omega$ on $\mathcal{U}$, normalized by $\omega (0) = 0$, satisfying the condition $\left \vert \omega (z)\right \vert < 1$. If $\omega \in \Omega$ and

then for any complex number $t$,

The above inequality is sharp for $\omega (z) = z$ or $\omega (z) = z^{2}$.

Lemma 2.3. Let $p(z) = 1+\sum \limits_{n = 1}^{\infty }p_{n}z^{n}\in \mathcal{P}[b, A, B],$ $b\in \mathbb{C}\backslash \{0\},$ $-1\leq B < A\leq 1.$ Then for any complex number $\mu$,

This result is sharp.

Proof. Let $p\in \mathcal{P}[b, A, B]$. Then we have

or, equivalently

which would further give

Comparing the coefficients of $z$ and $z^{2}$, we obtain

By a simple computation,

Now by using Lemma 2.2 with $t = \mu b(A-B)+B$, we get the required result. Equality holds for the functions

Now we prove the following result by using a method similar to the one in Libera ^[12].

Lemma 2.4. Suppose that $N$ and $D$ are analytic in $\mathcal{U}$ with $N(0) = D(0) = 0$ and $D$ maps $\mathcal{U}$ onto a many sheeted region which is starlike with respect to the origin. If $\frac{N^{\prime }(z)}{D^{\prime }(z) }\in \mathcal{P}[b, A, B]$, then

Proof. Let $\frac{N^{\prime }(z)}{D^{\prime }(z)}\in \mathcal{P}[b, A, B].$ Then by using a result due to Attiya ^[6], we have

where $c(r) = \frac{1-B\left[B+b\left(A-B\right) \right] r^{2}}{1-B^{2}r^{2}}$ and $d\left(r\right) = \frac{\left \vert b\right \vert \left(A-B\right) r^{2}}{1-B^{2}r^{2}}.$ We choose $A\left(z\right)$ such that $\left \vert A\left(z\right) \right \vert < d\left(r\right)$ and

Now for a fixed $z_{0}$ in $\mathcal{U}$, consider the line segment $L$ joining $0$ and $D\left(z_{0}\right)$ which remains in one sheet of the starlike image of $\mathcal{U}$ by $D.$ Suppose that $L^{-1}$ is the pre-image of $L$ under $D$. Then

This implies that

Therefore

For $A = -B = b = 1$, we have the following result due to Libera ^[12].

Lemma 2.5. If $N$ and $D$ are analytic in $\mathcal{U}$ with $N(0) = D(0) = 0$ and $D$ maps $\mathcal{U}$ onto a many sheeted region which is starlike with respect to the origin, then

Lemma 2.6. ^[14] If $-1\leq B < A\leq 1, \beta _{1} > 0$ and the complex number $\gamma$ satisfies ${Re}\left \{ \gamma \right \} \geq -\beta_{1}\left(1-A\right) /\left(1-B\right),$ then the differential equation

has a univalent solution in $\mathcal{U}$ given by

If $p\left(z\right) = 1+p_{1}z+p_{2}z^{2}+\cdots$ is analytic in $\mathcal{U}$ and satisfies

then

and $q\left(z\right)$ is the best dominant.

3.   Some auxiliary results

Before proving the results for the generalized Bazilevič functions, let us discuss a few results related to the function $g \in S^*[A, B]$.

Theorem 3.1. Let $g\in \mathcal{S}^{\ast }[A, B]$ and of the form $\left(1.3\right)$. Then for any complex number $\mu,$

Proof. The proof of the result is the same as of Lemma 2.3. The result is sharp and equality holds for the function defined by

or

Theorem 3.2. Let $g\in \mathcal{S}^{\ast }[A, B].$ Then for $c > 0,$ $\alpha > 0$ and $\beta$ any real number,

is in $\mathcal{S}^{\ast }[A, B]$. In addition

where

Proof. From $\left(3.1\right),$ we have

Differentiating and rearranging gives

where $p(z) = \frac{zG^{\prime }(z)}{G(z)}$. Then differentiating $\left(3.2\right)$ logarithmically, we have

Since $g\in \mathcal{S}^{\ast }[A, B]$, it follows that

Now by using Lemma 2.6, for $\beta _{1} = \alpha$ and $\gamma = c+i\beta,$ we obtain

where

Now by using the properties of the familiar hypergeometric functions proved in ^[15], we have

This implies that

and

Theorem 3.3. Let $g\in \mathcal{S}^{\ast }[A, B].$ Then

is $(\alpha +c)$-valent starlike, where $\alpha > 0,$ $c > 0$ and $\beta$ is a real number.

Proof. Let $D_{1}(z) = zS^{\prime }(z) = z^{c+i\beta }g^{\alpha }(z)$ and $N_{1}(z) = S(z).$ Then

Since $g\in \mathcal{S}^{\ast }[A, B]\subset \mathcal{S}^{\ast }\left(\frac{ 1-A}{1-B}\right)$, see ^[10], it follows that

Also

Now by using Lemma 2.5$,$ we have

By the mean value theorem for harmonic functions,

Therefore

Now by using a result due to [9,p 212], we have that $S$ is $(c+\alpha)$-valent starlike function.

4.   Main results

Now we are ready to discuss some results related to the defined generalized Bazilevič functions.

Theorem 4.1. Let $f$ be a generalized Bazilevič function associated by the quadruple $\left(\alpha, \beta, g, p\right),$ where $g\in \mathcal{S} ^{\ast }[A, B]$ of the form $\left(1.3\right)$ and $p\in \mathcal{P} [b, A, B]$ of the form $\left(1.2\right)$. Then for $c > 0,$

is a generalized Bazilevič function associated by the quadruple $(\alpha, \beta, G, p),$ where $G\in \mathcal{S}^{\ast }[A, B, \delta]$, as defined by $\left(3.1\right).$

Proof. From $\left(4.1\right)$, we have

This implies that

Differentiate both sides and rearrange, we get

and

Now from $(3.1)$, we have

With this, note that

which implies $\frac{N^{\prime }(z)}{D^{\prime }(z)} \in \mathcal{P}[b, A, B]$. By Theorem 3.3, we know that $D(z) = \int_{0}^{z}t^{c+i\beta -1}g^{\alpha }(t)dt$ is $(\alpha +c)$-valent starlike. Therefore by using Lemma 2.4, we obtain

This is the equivalent form of Definition 1.1. Hence the result follows.

Corollary 4.2. Let $A = 1, B = -1$ and $\beta = 0$ in Theorem 4.1. Then

belong to $S^{\ast }\left(\delta _{1}\right)$, where

Hence $G$ is starlike when $g\in \mathcal{S}^{\ast },$ and

belongs to the class of Bazilevič functions associated by the quadruple $(\alpha, 0, G, p).$

Theorem 4.3. Let $f$ of the form $\left(1.1\right)$ be a generalized Bazilevič function associated by the quadruple $(\alpha, \beta, g, p)$, with $g\in \mathcal{S}^{\ast }[A, B]$ of the form $\left(1.3\right)$ and $p\in \mathcal{P}[b, A, B]$ of the form $\left(1.2\right)$. Then

This inequality is sharp.

Proof. Since $f$ is a generalized Bazilevič function associated by the quadruple $(\alpha, \beta, g, p)$, we have

As $f, \ g$ and $p$ respectively have the form $\left(1.1\right), \left(1.3\right)$ and $\left(1.2\right),$ it is easy to get

Putting these values in $\left(4.2\right)$ and comparing the coefficients of $z,$ we obtain

Similarly by comparing the coefficients of $z^{2}$ and rearranging, we have

From $\left(4.4\right)$, we have

Now by using Theorem 3.1 and Lemma 2.3, both with $\mu = \frac{1}{2}$, we obtain

and

Therefore, we have

The equality

for $B\neq 0$ can be obtained for

Similarly, the equality

for $B = 0$ can be obtained for the function $g_{\ast }(z) =$ $ze^{\frac{A}{2} z^{2}} = z+\frac{A}{2}z^{3}+\cdots.$ Also equality for the functional $\left \vert p_{2}-\frac{1}{2}p_{1}^{2}\right \vert$ can be obtained by the functions

Corollary 4.4. For $A = 1,$ $B = -1$ and $b = 1$, we have the result proved in ^[8]:

For $\alpha = 1$, $\beta = 0$, we have $f\in \mathcal{K}$, the class of close-to-convex functions, and

The latter result has been proved in ^[11].

Theorem 4.5. Let $f$ of the form $\left(1.1\right)$ be a generalized Bazilevič function associated by the quadruple $(\alpha, \beta, g, p)$, with $g\in \mathcal{S}^{\ast }[A, B]$ and of the form $\left(1.3\right)$ and $p\in \mathcal{P}[b, A, B]$ of the form $\left(1.2\right)$. Then

(i)

(ii) If $\alpha = 0,$ then

Both the above inequalities are sharp.

Proof. (ⅰ) From $\left(4.3\right)$, we have

This implies that

By using the coefficient bound for $\mathcal{S}^{\ast }[A, B]$ along with the coefficient bound of $\mathcal{P}[b, A, B]$, we have

This implies that

Equality can be obtained by the functions

(ⅱ) Let $\alpha = 0.$ Then from $\left(4.3\right)$ and $\left(4.4 \right)$, we have

This implies

where $\mu = \frac{1}{2}-\frac{(3+i\beta)}{2\left(1+i\beta \right) ^{2}}.$ Now by using Lemma 2.3, we obtain

Sharpness can be attained by the functions

Corollary 4.6. For $A = 1,$ $B = -1$ and $b = 1$, we have

and

In the final part of this paper, we look at some results for the generalized Bazilevič functions associated with $\beta = 0$.

Let $C_{r}$ denote the closed curve which is the image of the circle $\left \vert z\right \vert = r < 1$ under the mapping $w =$ $f(z)$, and $L_{r}\left(f(z)\right)$ denote the length of $C_{r}$. Also let $M(r) = {\max} _{\left \vert z\right \vert = r} \left \vert f(z)\right \vert \ $and $m(r) = { \min}_{\left \vert z\right \vert = r} \left \vert f(z)\right \vert$. We now prove the following result.

Theorem 4.7. Let $f$ be a generalized Bazilevič function associated by the quadruple $(\alpha, 0, g, p)$. Then for $B\neq 0$,

where

Proof. As $f$ is a generalized Bazilevič function associated by the quadruple $(\alpha, 0, g, p)$, we have

where $g\in \mathcal{S}^{\ast }[A, B]$ and $p\in \mathcal{P}[b, A, B]$. Since for $z = re^{i\theta }$, $0 < r < 1$,

we have for $0 < \alpha \leq 1$,

where $\frac{zg^{\prime }(z)}{g(z)} = h\left(z\right) \in \mathcal{P}[A, B]$. Now by using the distortion theorem for Janowski starlike functions when $B\neq 0$ (see ^[10]) and the Cauchy-Schwarz inequality, we have

Now by using Lemma 2.1 for both the classes $\mathcal{P}[b, A, B]$ and $\mathcal{P}[A, B]$, along with the result

for $p\in \mathcal{P}[b, A, B]$ (see ^[19]), we can write

Since $1-\left \vert B\right \vert r\geq 1-r$ and $1- B^2 r^2\geq 1-r^2$,

where $C(\alpha, b, A, B) = 2\pi |b| [(A-B) + (1/ \alpha)] B$.

When $\alpha > 1,$ we can prove similarly as above to get

Corollary 4.8. For $g\in \mathcal{S}^{\ast }$ and $p\in \mathcal{P}(b),$ we have

Theorem 4.9. Let $f$ be a generalized Bazilevič function associated by the quadruple $(\alpha, 0, g, p)$, where $g\in \mathcal{S}^{\ast }[A, B]$ and $p\in \mathcal{P} [b, A, B]$. Then for $B\neq 0$,

Proof. By Cauchy's theorem for $z = re^{i\theta },$ $n\geq 2,$ we have

Therefore

By using Theorem 4.7 for the case $0 < \alpha \leq 1,$ we have

Hence, by taking $r$ approaches $1^{-}$,

For $\alpha > 1,$ we have

Theorem 4.10. Let $f$ be a generalized Bazilevič function represented by the quadruple $(\alpha, 0, g, p),$ where $g\in \mathcal{S}^{\ast }[A, B]$ and $p\in \mathcal{P }[b, A, B].$ Then for $B\neq 0$,

Proof. Since $f$ is a generalized Bazilevič function associated by the quadruple $(\alpha, 0, g, p),$ by definition, we have

where $g\in \mathcal{S}^{\ast }[A, B]$ and $p\in \mathcal{P}[b, A, B].$ This implies that

and so

Now by using the results

and

we have

Putting $s = ru,$ we have

where ${_{2}}F_{1} \left(a; b;c; z\right)$ is the hypergeometric function.

Corollary 4.11. For $g\in \mathcal{S}^{\ast \mathit{\text{}}}$and $p\in \mathcal{P},$ we have

Acknowledgments

The research for the fourth author is supported by the USM Research University Individual Grant (RUI) 1001/PMATHS/8011038.

Conflict of interest

The authors declare that they have no conflict of interests.