Bounded Turning of an m-th Partial Sum of Modified Caputo’s Fractional Calculus Derivative Operator
Abstract: In this article, we consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk, we investigate certain conditions under which the partial sums of the modi?ed Caputo’s fractional derivative operators of analytic univalent functions of bounded turning are also of bounded turning. 1. Introduction and Deﬁnitions

Let $\mathcal{A}$ denote a class of all analytic functions of the form

$f\left(z\right)=z+\underset{k=2}{\overset{\infty }{\sum }}{a}_{k}{z}^{k}$ (1.1)

which are analytic in the open unit disk $U=\left\{z:|z|<1\right\}$ and normalized by $f\left(0\right)={f}^{\prime }\left(0\right)-1=0$

Deﬁnition 1.

Let $B\left(\mu \right),0\le \mu <1$ denote the class of functions of the Form (1.1) then if $\Re \left\{{f}^{\prime }\right\}>\mu$ , that is the real part of its first derivative map the unit disk onto the right half plane, then the class of functions in $B\left(\mu \right)$ are called functions of bounded turning.

By Nashiro Warschowski, see  , it is proved that the functions in $B\left(\mu \right)$ are univalent and also close to convex in U. In  , it was also shown that the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning. For more works on bounded turning see   .

Deﬁnition 2.

If $f\left(z\right)=\underset{n=0}{\overset{\infty }{\sum }}{a}_{n}{z}^{n}$ and $g\left(z\right)=\underset{n=0}{\overset{\infty }{\sum }}{b}_{n}{z}^{n}$ are analytic in U, then their Hadamard product $f*g$ defined by the power series is given by:

$\left(f*g\right)\left(z\right)=\underset{n=0}{\overset{\infty }{\sum }}{a}_{n}{b}_{n}{z}^{n}.$ (1.2)

Note that the convolution so defined is also analytic in U.

For ƒ of the Form (1.1) several interesting derivatives operators in their different forms have been studied, here we consider (1.1) using the modified Caputo’s derivative operator ${J}_{\eta ,\lambda }f\left(z\right)$ , see   , stated as follow:

For $f\in \mathcal{A}$ , ${J}_{\eta ,\lambda }f\left(z\right)=\frac{2+\eta -\lambda }{\eta -\lambda }{z}^{\lambda -\eta }{\int }_{0}^{z}\frac{{\Omega }^{\eta }f\left(\xi \right)}{{\left(z-\xi \right)}^{\lambda +1-\eta }}\text{d}\xi$ (1.3)

where $\eta$ is a real number and $\eta -1<\lambda \le \eta <2$ . Notice that (1.3) can also be express as:

${J}_{\eta ,\lambda }f\left(z\right)=z+\underset{n=2}{\overset{\infty }{\sum }}\frac{{\left(n+1\right)}^{2}\left(2+\eta -\lambda \right)\left(2-\eta \right)}{\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)}{a}_{n}{z}^{n}$ (1.4)

and its partial sum given as:

${P}_{M}\left(z\right)=z+\underset{n=2}{\overset{M}{\sum }}\frac{{\left(n+1\right)}^{2}\left(2+\eta -\lambda \right)\left(2-\eta \right)}{\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)}{a}_{n}{z}^{n}$ (1.5)

We determine conditions under which the partial sums of the operator given in (1.4) are of bounded turning. We shall use the following lemmas in the sequel to establish our result.

Lemma 1. 

For $z\in U$ , we have

$\Re \left\{\underset{n=1}{\overset{\infty }{\sum }}\frac{{z}^{n}}{n+2}\right\}>-\frac{1}{3},\left(z\in U\right)$ (1.6)

Lemma 2. 

Let P(z) be analytic in U, such that P(0) = 1, and $\Re \left(P\left(z\right)\right)>\frac{1}{2}$ in U. For

function Q analytic in U the convolution function $P*Q$ takes values in the convex hull of the image U under Q.

We shall implore lemmas 1 and 2 to show conditions under which the m-th partial sum (2.1) of the modiﬁed Caputoes derivative operator of analytic univalent functions of bounded turning is also of bounded turning.

2. Main Theorem

Let $f\left(z\right)\in \mathcal{A}$ be of the Form (1.1), if $\frac{1}{2}<\mu <1$ and $f\left(z\right)\in B\left(\mu \right)$ , then ${P}_{M}\left(z\right)\in B\left(\frac{\left(3-\left(2+\eta -\lambda \right)\left(2-\eta \right)\left(1-\mu \right)\right)}{3}\right)$ , $\eta -1<\lambda \le \eta <2$ .

Proof.

Let $f\left(z\right)$ be of the Form (1.1) and $\Re \left\{{f}^{\prime }\left(z\right)\right\}>\mu ,\frac{1}{2}<\mu <1,z\in U$ . This implies that

$\Re \left\{1+\underset{n=2}{\overset{\infty }{\sum }}n{a}_{n}{z}^{n-1}\right\}>\frac{\mu }{2}$ (2.1)

Now for $\frac{1}{2}<\mu <1$ we have

$\Re \left\{1+\underset{n=2}{\overset{\infty }{\sum }}{a}_{n}\frac{n}{1-\mu }{z}^{n-1}\right\}>\Re \left\{1+\underset{n=2}{\overset{\infty }{\sum }}n{a}_{n}{z}^{n-1}\right\}$ (2.2)

Applying the convolution properties to ${P}^{\prime }\left(z\right)$ , where

${{P}^{\prime }}_{M}\left(z\right)=1+\underset{n=2}{\overset{M}{\sum }}\frac{n{\left(n+1\right)}^{2}\left(2+\eta -\lambda \right)\left(2-\eta \right)}{\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)}{a}_{n}{z}^{n-1}$ (2.3)

$\begin{array}{l}\left[\left\{1+\underset{n=2}{\overset{\infty }{\sum }}{a}_{n}\frac{n}{1-\mu }{z}^{n-1}\right\}\right]*\left[1+\underset{n=2}{\overset{M}{\sum }}\frac{n{\left(n+1\right)}^{2}\left(2+\eta -\lambda \right)\left(2-\eta \right)}{\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)}\left(1-\mu \right){a}_{n}{z}^{n-1}\right]\\ =P\left(z\right)*Q\left(z\right)\end{array}$ (2.4)

with recourse for Lemma 1 and $J=m-1$ we have

$\Re \left\{\underset{n=2}{\overset{M}{\sum }}\frac{{z}^{n-1}}{n+1}\right\}>-\frac{1}{3}$ (2.5)

Then for $\eta -1<\lambda \le \eta <2$

$\begin{array}{l}\Re \left\{\underset{n=2}{\overset{M}{\sum }}\frac{{z}^{n-1}}{{\left(n{\left(n+1\right)}^{2}\right)}^{-1}\left(2+\eta -\lambda \right)\left(2-\eta \right)\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)\left(1-\mu \right){a}_{n}{z}^{n-1}}\right\}\\ \ge \Re \left\{\underset{n=2}{\overset{M}{\sum }}\frac{{z}^{n-1}}{n+1}\right\}\end{array}$ (2.6)

Hence

$\begin{array}{l}\Re \left\{\underset{n=2}{\overset{M}{\sum }}\frac{{z}^{n-1}}{{\left(n{\left(n+1\right)}^{2}\right)}^{-1}\left(2+\eta -\lambda \right)\left(2-\eta \right)\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)\left(1-\mu \right){a}_{n}{z}^{n-1}}\right\}\\ \ge -\frac{1}{3}\end{array}$ (2.7)

Relating Lemma 1 and with $Q\left(z\right)$ , a computation gives

$\begin{array}{c}\Re Q\left(z\right)=\left\{1+\underset{n=2}{\overset{M}{\sum }}\frac{n{\left(n+1\right)}^{2}\left(2+\eta -\lambda \right)\left(2-\eta \right)}{\left(n+\eta -\lambda +1\right)\left(n-\eta +1\right)}\left(1-\mu \right){a}_{n}{z}^{n-1}\right\}\\ >\frac{3-\left(2+\eta -\lambda \right)\left(2-\eta \right)\left(1-\mu \right)}{3}\end{array}$ (2.8)

Recall the power series

$P\left(z\right)=\left\{1+\underset{n=2}{\overset{\infty }{\sum }}{a}_{n}\frac{n}{1-\mu }{z}^{n-1}\right\},z\in U$ (2.9)

satisfies $p\left(0\right)=1$ and $\Re \left(P\left(z\right)\right)=\Re \left\{1+\underset{n=2}{\overset{\infty }{\sum }}{a}_{n}\frac{n}{1-\mu }{z}^{n-1}\right\}>\frac{1}{2},z\in U$ . Therefore by Lemma 2 we have

$\Re \left({P}^{\prime }\left(z\right)\right)>\frac{3-\left(2+\eta -\lambda \right)\left(2-\eta \right)\left(1-\mu \right)}{3},z\in U$ (2.10)

This proves our results.

Cite this paper: Terwase, A. , Longwap, S. and Choji, N. (2020) Bounded Turning of an m-th Partial Sum of Modified Caputo’s Fractional Calculus Derivative Operator. Open Access Library Journal, 7, 1-4. doi: 10.4236/oalib.1105324.
References

   Goodman, A.W. (1983) Univalent Functions. Vol. I & II. Polygonal Publishing House, Washinton, New Jersey.

   Jahangiri, J.M. and Farahmand, K. (2004) Partial Sums of Functions of Bounded Turning. International Journal of Mathematics and Mathematical Sciences, 2004, Article ID: 451028.
https://doi.org/10.1155/S0161171204305284

   Liu, J.L. and Owa, S. (1997) On Partial Sums of the Libera Integral Operator. Journal of Mathematical Analysis and Applications, 213, 444-454. https://doi.org/10.1006/jmaa.1997.5549

   Darus, M. and Ibrahim, R.W. (2010) Partial Sums of Analytic Functions of Bounded Turning with Applications. Computational & Applied Mathematics, 29, 81-88.
https://doi.org/10.1590/S1807-03022010000100006

   Jamal, S. (2016) A Note on the Modi?ed Caputos Fractional Calculus Derivative Operator. Far East Journal of Mathematical Sciences (FJMS), 100, 609-615.
https://doi.org/10.17654/MS100040609

   Salah, J. and Darus, M. (2010) A Subclass of Uniformly Convex Functions Associated with a Fractional Calculus Operator Involving Caputos Fractional Di?erentiation. Acta Universitatis Apulensis. Mathematics-Informatics, 24, 295-306.

   Gasper, G. (1969) Nonnegative Sums of Cosines, Ultraspherical and Jacobi Polynomials. Journal of Mathematical Analysis and Applications, 26, 60-68.
https://doi.org/10.1016/0022-247X(69)90176-0

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