Subclass of Bazilevič functions of complex order

1.   Introduction and preliminaries

Let $A$ be the class of functions $f$ of the form

which are analytic in the open unit disk $E = \left\{ z:\left\vert z\right\vert < 1\right\}$. If $f\left(z\right) \ $and $g\left(z\right)$ are analytic in $E, \ $we say $f\left(z\right) \ $is subordinate to $g\left(z\right), \ $written $f\prec g\ $or $f\left(z\right) \prec g\left(z\right).\ $If there exists a Schwarz function $w\left(z\right), \ w\left(0\right) = 0\ $and $\left\vert w\left(z\right) \right\vert < 1$ in $E\ $then $f\left(z\right) = g\left(w\left(z\right) \right).\ $Let $P\left(b\right),$ $b\neq 0\ $(complex) denote the class of analytic functions

such that $1+\frac{1}{b}\left\{ p\left(z\right) -1\right\} \in P, \ $where $P\ $is the well-known class of analytic functions with positive real part. The class $P\left(b\right) \ $is defined by Nasr and Aouf ^[15]$.$

Let $S^{\ast }\left(\gamma \right), \ 0\leq \gamma < 1\ $is the class of starlike univalent functions

of order $\gamma \ $such that ${Re}\frac{zg^{\prime }\left(z\right) }{ g\left(z\right) } > \gamma, \ z\in E.\ $This class was introduced by Robertson, for details, see ^[8].

The class of Bazilevič functions in the open unit disc $E\ $was first introduced by Bazilevič ^[3]$\ $in 1955. He defined Bazilevič functions by the relation

where $p\in P, \ g\in S^{\ast }, \ \alpha > 0\ $and $\beta \ $is any real. The class of Bazilevič function is the largest family of univalent function. Bazilevič showed that the class of Bazilevič function is univalent in $E$. Except this, a very little is known regarding the family as a whole. Indeed, it is easy to verify that, with special choices of the parameters and and the function $g(z)$, the class of Bazilevič functions reduces to some well-known subclasses of univalent functions. By choosing $g(z) = z$ and $\beta = 0,$ Singh ^[21] studied the class $B_{1}\left(\alpha \right)$ of Bazilevič functions. Recently, some authors have find coefficient bounds for this class of functions. In particular, Cho et al. ^[6] have studied coefficient difference for the class of Bazilevič functions. Fifth and sixth coefficient bounds for the subclass $B_{1}\left(\alpha \right)$ have been found by Cho and Kumar ^[5], and Marjono et al. ^[14]. For some more work, see ^{[1,7,9,11,12,19,20]}. In 1979 Campbell and Pearce ^[4]generalized the class of Bazilevi č functions by means of differential equation

They associate each generalized Bazilevič function $f\ $with the quadruple $\left(\alpha, \beta, g, p\right),$ where $g\in S^{\ast }\ $and $p\in P, \ \alpha > 0$ and $\beta \ $any real.

Definition 1.1. Let $S^{\ast }\left(\gamma \right)$ be the class of functions $g\ $of the form $\left(1.3\right)$ and let $P\left(b\right), \ b\neq 0$ (complex) be the class of normalized$\ $functions $p\ $defined by $\left(1.2\right).\ $Then a function $f\ $of the form $\left(1.1\right),$ analytic in $E, \ $belongs to the generalized Bazilevič functions associated with the quadruple $\left(\alpha, \beta, g, p\right) \ $if and only if $f\ $satisfies $\left(1.5\right)$ or

The above differential equation can be written as

Since $p\in P\left(b\right), \ $therefore we can write

where $g\in S^{\ast }\left(\gamma \right), \ 0\leq \gamma < 1, \ \alpha > 0\ $ and$\ \beta \ $is any real.

We have the following special cases.

(ⅰ) For $\gamma = 0, \ $we have the class of Bazilevič functions of complex order, defined by Noor ^[16].

(ⅱ) For $\gamma = 0, \ b = 1, \ $we obtain the generalized Bazilevič functions defined in ^[4].

In this paper, we study the class of functions $\left(\alpha, \beta, g, p\right)$. We study coefficient bounds, inclusion result, arc length problem and radii problems. Our results generalize some previously proven results.

We need the following lemmas which will be used in our main results.

Lemma 1.1. ^[8] Let $g\in S^{\ast }\left(\gamma \right),$ $0\leq \gamma < 1.$ Then

$\left(i\right)$ $\ \ \left\vert b_{n}\right\vert \leq \frac{1}{\left(n-1\right)!}\overset{n}{\underset{k = 2}{\prod\limits }}\left(k-2\gamma \right).$

$\left(ii\right)$ $\ \ \frac{r}{\left(1+r\right) ^{2\left(1-\gamma \right) }}\leq \left\vert g\left(z\right) \right\vert \leq \frac{r}{\left(1-r\right) ^{2\left(1-\gamma \right) }}, \ z = re^{i\theta }.$

These inequalities are sharp for the function $g_{0}\left(z\right) = \frac{z }{\left(1-z\right) ^{2\left(1-\gamma \right) }}.$

Lemma 1.2. Let $p\in P\left(b\right)$. Then, for $z = re^{i\theta }$

$\left(i\right) \ \ \ \frac{1}{2\pi }\overset{2\pi }{\underset{0}{\int }} \left\vert p\left(re^{i\theta }\right) \right\vert ^{2}d\theta \leq \frac{ 1+\left(4\left\vert b\right\vert ^{2}-1\right) r^{2}}{1-r^{2}}, \ \ \ $see ^[18]$,$

$\left(ii\right) \ \ \ \frac{1-2\left\vert b\right\vert r+\left(2{Re\ } b-1\right) r^{2}}{1-r^{2}}\leq \left\vert p\left(z\right) \right\vert \leq \frac{1+2\left\vert b\right\vert r+\left(2{Re\ }b-1\right) r^{2}}{1-r^{2}}.$

This result is the special case of the one, proved in ^[2].

Now we introduce the Hypergeometric function. Let $a_{1},$ $b_{1}$ and $c_{1}$ be complex numbers with $c_{1}\neq 0, -1, -2, \cdots.\ $The function

called, the confluent Gaussian hypergeometric, is analytic in $E$ and satisfies hypergeometric differential equation

This can be written as

Some properties of Gaussian hypergeometric are given in the following lemma.

Lemma 1.3. ^[22] Let $a_{1}, \ b_{1}\ $and $c_{1}\neq 0, -1, -2\cdots \ $be complex numbers. Then, for ${Re\ }c_{1} > {Re\ }b_{1} > 0$

Lemma 1.4. [10,inequality 7,p.10] Let $\Omega$ be the class of analytic functions $w, \ $normalized by $w\left(0\right) = 0, \ $satisfying the condition $\left\vert w\left(z\right) \right\vert < 1.\ $If $w\in \Omega \ $and $w\left(z\right) = w_{1}z+w_{2}z^{2}+\cdots, \ z\in E, \ $then

for any complex number $t.\ $The result is sharp for the functions $w\left(z\right) = z^{2}\ $or $w\left(z\right) = z.$

Lemma 1.5. Let $p\in P\left(b\right), \ b\neq 0\ $(complex) and of the form $\left(1.2\right)$. Then for $\mu \ $a complex number

This result is sharp.

Proof. Since $p\in P\left(b\right), \ $therefore we can write

Thus

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

Now using Lemma 1.4 we have the required result. This result is sharp for the functions

or

Lemma 1.6. Let $g\in S^{\ast }\left(\gamma \right), \ 0\leq \gamma < 1\ $and of the form $\left(1.3\right).\ $Then for $\mu \ $complex

This result is best possible.

This result is a special case of the one, proved in ^[10].

Lemma 1.7. ^[16] If $N\ $and $D\ $are analytic in $E, \ N\left(0\right) = D\left(0\right) = 0, \ D\ $maps $E\ $onto a many sheeted region which is starlike with respect to the origin, then

Lemma 1.8. [13,p. 109] Let $\beta _{1}, \ \gamma _{1}, \ A\in \mathbb{C}, \ $with ${Re}\left[ \beta _{1}+\gamma _{1}\right] > 0, \ $and let $B\in \left[ -1, 0\right] \ $satisfy

or

If $h\left(z\right) = 1+\underset{n = 1}{\overset{\infty }{\sum\limits }}c_{n}z^{n}\ $ satisfies

then $h\left(z\right) \prec q\left(z\right) \prec \frac{1+Az}{1+Bz}, \ $ where $q\left(z\right) \ $is the best dominant and

with

Lemma 1.9. Let $g\in S^{\ast }\left(\gamma \right), \ 0\leq \gamma < 1.\ $ Then

$c > 0, \ \alpha > 0\ $and $\beta \ $any real, belongs to $S^{\ast }\left(\delta \right), \ $where $\delta = \underset{\left\vert z\right\vert = 1}{ \min }{Re}q\left(z\right) \ $and

Proof. From $\left(1.9\right), \ $we have

where $\frac{zG^{\prime }\left(z\right) }{G\left(z\right) } = p\left(z\right).\ $Differentiating $\left(1.10\right) \ $logarithmically and using the fact that $g\in S^{\ast }\left(\gamma \right), \ $we have

Now using the Lemma 1.8 for $A = 1-2\gamma, \ B = -1, \ \beta _{1} = \alpha,$ $\gamma _{1} = c+i\beta \ $and then Lemma 1.3 we have the required result $.$

Throughout the main results we assume that $g\ $belongs to the class of starlike functions of order $\gamma \ $and $p\in P\left(b\right) \ $unless otherwise stated.

2.   Main results

Theorem 2.1. Let the generalized Bazilevič function $f$ be represented by the quadruple $\left(\alpha, 0, g, p\right).$ Then, for $\alpha > 0$

where $_{2}F_{1}\ $is the Gauss hypergeometric function.

Proof. Since $f$ is represented by $\left(\alpha, 0, g, h\right), \ $therefore from $\left(1.6\right), \ $we have

This implies that

Thus

Using the Lemma $1.1\left(ii\right)$ and Lemma$\ 1.2\left(ii\right), \ $we obtain

Now for $t = ru\ $and using Lemma 1.3, we have

Hence the proof is completed.

Theorem 2.2. Let $f\ $be generalized Bazilevič function represented by the quadruple $\left(\alpha, 0, g, p\right).\ $Then

where $m\left(r\right) = \underset{\left\vert z\right\vert = r}{\min } \left\vert f\left(z\right) \right\vert, \ M\left(r\right) = \underset{ \left\vert z\right\vert = r}{\max }\left\vert f\left(z\right) \right\vert \ $ and $C\left(b\right)$ is constant depending upon $b\ $only.

Proof. We know that

Since $f$ is generalized Bazilevič function represented by the quadruple $\left(\alpha, 0, g, h\right),$ therefore

This implies that

Now using Cauchy Schwarz inequality, we have

By Lemma 1.1$\left(ii\right), \ $Lemma 1.2$\left(i\right) \ $and a subordination result, we obtain

Similarly for $\alpha > 1, \ $we have

For $\gamma = 1, \ $we have the following result, proved by Noor$\ $^[16]$.$

Corollary 2.3. Let $g\in S^{\ast }\ $and $p\in P\left(b\right).\ $Then, for $0 < \alpha \leq 1$

Theorem 2.4. Let $f\ $be generalized Bazilevič function represented by the quadruple $\left(\alpha, 0, g, p\right).\ $Then

where $m, \ M\ $are the same as in Theorem 2.2$\ $and $C_{1}\left(b\right)$ is a constant depending upon $b\ $only.

Proof. Since with $z = re^{i\theta },$ Cauchy theorem gives

Therefore

Now using Theorem 2.2 for $0 < \alpha \leq 1, \ $we have

Putting $r = 1-\frac{1}{n}, \ $we have

Similarly we have for $\alpha > 1.$

For $\gamma = 1, \ $we have the following result, proved by Noor$\ $^[16]$.$

Corollary 2.5. Let $g\in S^{\ast }\ $and $p\in P\left(b\right).\ $Then, for $0 < \alpha \leq 1$

Theorem 2.6. Let the generalized Bazilevič function$\ $be represented by the quadruple $\left(\alpha, \beta, g, p\right).\ $Then

This result is best possible.

Proof. Since $f$ is generalized Bazilevič function, therefore we have

Multiplying both sides by $f\left(z\right) f^{\prime }\left(z\right) g\left(z\right) p\left(z\right), \ $we obtain

Since $f\left(z\right) = z+\underset{n = 2}{\overset{\infty }{\sum\limits }} a_{n}z^{n}, \ g\left(z\right) = z+\underset{n = 2}{\overset{\infty }{\sum\limits }} b_{n}z^{n}\ $and $p\left(z\right) = 1+\underset{n = 1}{\overset{\infty }{\sum\limits }}p_{n}z^{n}, \ $therefore comparing the coefficients of $z^{3}, \ $we have

Thus

Similarly comparing the coefficients of $z^{4}\ $and using above inequality, we obtain

From $\left(2.3\right) \ $and $\left(2.4\right)$, we obtain

Now using Lemma 1.5 and Lemma 1.6 for $\mu = \frac{1}{2}, \ $we have

Result is sharp for the function $f_{0}$ represented by the quadruple $\left(\alpha, \beta, \frac{z}{\left(1-z\right) ^{2\left(1-\gamma \right) } }, \frac{1+\left(2b-1\right) z}{1-z}\right)$ or $f_{1}$ represented by the quadruple $\left(\alpha, \beta, \frac{z}{\left(1-z\right) ^{2\left(1-\gamma \right) }}, \frac{1+\left(2b-1\right) z^{2}}{1-z^{2}}\right).$

For $\gamma = 0, \ b = 1, \ $we have the result proved by Campbell and Pearce ^[4].

Corollary 2.7. Let $g\in S^{\ast }\ $and $p\in P.\ $Then

Theorem 2.8. (i) If $f\ $is generalized Bazilevič function with representation $\left(\alpha, \beta, g, p\right),$ then for$\ \alpha \geq 0$

(ii) If $f\ $is in $\left(0, \beta, g, p\right), \ $then

Both the inequalities are sharp.

Proof. (ⅰ) From $\left(2.3\right), \ $we have

This implies that

Using Lemma 1.1$\left(i\right), \ $we have $\left\vert b_{2}\right\vert \leq 2\left(1-\gamma \right) \ $and using the fact that$\ \left\vert p_{1}\right\vert \leq 2\left\vert b\right\vert, \ $we obtain

Result is sharp for the functions $f_{0}$ defined by the quadruple $\left(\alpha, \beta, \frac{z}{\left(1-z\right) ^{2\left(1-\gamma \right) }}, \frac{1+\left(2b-1\right) z}{1-z}\right)$.

(ⅱ) Since $f\ $is represented by the quadruple $\left(0, \beta, g, p\right), \ $therefore $\left(2.3\right) \ $and $\left(2.4\right) \ $yield

and

Therefore

Using Lemma 1.5 for $\mu = \frac{1}{2}\left(1-\frac{3+i\beta }{\left(1+i\beta \right) ^{2}}\right), \ $we have the required result. This result is best possible for the function$\ f_{0}\ $represented by the quadruple $\left(0, \beta, g, \frac{1+\left(2b-1\right) z}{1-z}\right)$ or $f_{1}$ represented by the quadruple $\left(0, \beta, g, \frac{1+\left(2b-1\right) z^{2}}{1-z^{2}}\right).$

For $\gamma = 0\ $and $b = 1, \ $we have the following result proved in ^[4]$.$

Corollary 2.9. Let $g\in S^{\ast }\ $and $p\in P.\ $Then

and for $\alpha = 0$

Theorem 2.10. Let $f\ $be a generalized Bazilevič function represented by $\left(\alpha, \beta, g, p\right).$ Then

belongs to the class of generalized Bazilevič functions represented by $\left(\alpha, \beta, G, p\right),$ where $G\in S^{\ast }\left(\delta \right), \ $defined by $\left(1.9\right) \ $with $c > 0, \ \alpha > 0\ $and $\beta \ $is any real.

Proof. From $\left(2.5\right), \ $we have

This implies that

Using$\ \left(1.9\right), \ $we obtain

Therefore

Since $D\left(z\right) = \underset{0}{\overset{z}{\int }}t^{c+i\beta -1}g^{\alpha }\left(t\right) dt\ $is $\left(\alpha +c\right) \ $valent starlike function therefore using Lemma 1.7, we have the required result.

Corollary 2.11. Let$\ \beta = 0\ $and $\gamma = 0.\ $Then

belongs to $S^{\ast }\left(\delta _{1}\right), \ $where$\ \delta _{1} = \frac{ -\left(1+2c\right) +\sqrt{\left(1+2c\right) ^{2}+8\alpha }}{4\alpha }.\ $ Hence $G\left(z\right) \ $is starlike, when $g\in S^{\ast },$therefore

belongs to the class of Bazilevič function represented by the quadruple $\left(\alpha, 0, g, p\right).$ This result is proved by Noor ^[16]$.$

Theorem 2.12. Let $f\ $is generalized Bazilevič function represented by the quadruple $\left(\alpha, 0, g, p\right).\ $Then $f\ $is $\frac{1}{\alpha }$-convex for $r_{0}\in \left(0, 1\right),$ where $r_{0}\ $is the least positive root of the equation

where

Proof. Since $\ f\ $is generalized Bazilevič function, therefore

where $p_{1}\in P\left(\gamma \right), \ 0\leq \gamma < 1\ $and $p\in P\left(b\right), \ b\neq 0\ $(complex). Therefore

Using Lemma 1.2$\left(ii\right) \ $and well-known distortion result for the class $P\left(\gamma \right), \ $we have

Since $\alpha \left(1-r^{2}\right) \left(1+2\left\vert b\right\vert r+\left(2{Re\ }b-1\right) r^{2}\right) > 0, \ $for ${Re\ }b\geq 1, \ $ therefore we must have $Ar^{4}+Br^{3}+Cr^{2}+Dr+\alpha > 0.$ Now $Q\left(0\right) = \alpha > 0\ $and $Q\left(1\right) = -4\left({Re\ }b+\left\vert b\right\vert \right) < 0.\ $Hence $f\ $is $\frac{1}{\alpha }$-convex.

The following result is proved in ^[17].

Corollary 2.13. For $b = 1, \gamma = 0$ and $\alpha > 0, \ f\ $is $\frac{1}{\alpha }$-convex for $r_{0} = \frac{\left(\alpha +1\right) -\sqrt{2\alpha +1}}{\alpha }.$

3.   Conclusion

In this paper, we have generalized the class of Bazilevi č functions associated with the quadruple $\left(\alpha, \beta, g, p\right)$ by taking the generalized versions of functions $g$ and $p$. This generalization unifies and generalizes certain already known classes of Bazilevič functions. We have explored certain aspects of this generalized class which includes coefficient bounds, radius problem, inclusion of Bernardi integral operator and arc length problem. Our results generalize various results in the literature.

There is still more to explore about these functions which includes coefficient bounds, Hankel determinants and Toeplitz determinants with multiple orders. Moreover, several generalizations can also be introduced and studied by suitable variation in quadruple $\left(\alpha, \beta, g, p\right)$. In particular, the assumption of generalized versions of starlike and caratheodory functions can result the proposed generalizations.

Acknowledgments

Authors are thankful to the anonymous referees for their valuable comments and suggestions.

Conflict of interest

Authors declare that they have no conflict of interest.