# Virendra KumarRAMANUJAN COLLEGE · Mathematics

Virendra Kumar

Post Doc, Ph.D. (Mathematics)

Presently working on differential subordination, estimation of Hankel and Hermitian-Toeplitz determinants!

## About

43

Publications

16,157

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

521

Citations

Citations since 2017

Introduction

Currently working as an Assistant Professor at Department of Mathematics, Ramanujan College, University of Delhi. Earlier, I worked as a Researcher, Department of Applied Mathematics, Pukyong National University, Busan, South, Korea during March 2017 to March 2018. Prior to this, I worked as an Assistant Professor & In-Charge, Department of Mathematics. I did Ph. D. (Mathematics, Complex Analysis) from Delhi Technology University (Formerly Delhi College of Engineering), Delhi, India (Feb 2010- May 2015). My field of research interest is the Geometric Function Theory.

**Skills and Expertise**

Additional affiliations

March 2017 - March 2018

July 2015 - March 2017

August 2008 - January 2010

Education

February 2010 - May 2015

## Publications

Publications (43)

Several properties of the class S∗r(α) of starlike functions of reciprocal order α (0 ≤ α < 1)
defined on the open unit disk have been studied in this paper. The paper begins with a sufficient condition for analytic functions to be in the class S∗r(α). Further, the sharp bounds on third order HermitianToeplitz determinant, initial inverse coefficie...

Sharp bounds on the moduli difference of successive inverse and logarithmic coefficients for a class of strongly Ozaki close-to-convex functions are investigated. Relevant connections of the main results presented in this paper with existing ones are also pointed out.

In this paper, sharp lower and upper bounds on the third order Hermitian-Toeplitz determinant for the classes of Sakaguchi functions and some of its subclasses related to right-half of lemniscate of Bernoulli, reverse lemniscate of Bernoulli and exponential functions are investigated.

The Littlewood-Paley conjecture hardly holds for any subclass of univalent functions except the class of starlike functions as verified, in general, by the researchers until now. So it is interesting to consider the classes where the Littlewood-Paley conjecture holds completely or partially. For such investigation, the classes of normalized strongl...

Sharp bounds on the second and third order Hermitian-Toeplitz determinants, initial logarithmic and inverse coefficients for functions in the class of non-Bazilevič functions are determined.

In this paper, we answer the questions raised in the paper [On the difference of inverse coefficients of univalent functions, Symmetry, 2020, 12(12), art. 2040, 14pp] by Sim and Thomas, and aim to verify the conjecture posed therein in certain cases. For this purpose, we investigate sharp bounds on moduli difference of successive inverse coefficien...

There is a rich literature on estimation of second and third Hankel determinants for normalised analytic functions in geometric function theory. It is also, therefore, natural to explore the concept of the Hermitian–Toeplitz determinants for such functions. In this paper, the sharp lower and upper estimations for third-order Hermitian-Toeplitz dete...

The sharp upper and lower bounds on the Hermitian-Toeplitz determinant of third order are computed for the classes of strongly starlike functions, lemniscate starlike functions and lune starlike functions. Moreover, a non-sharp upper bound on the fourth Hankel determinant for the lemniscate starlike functions is also obtained. Relevant connections...

The sharp lower and upper estimates on the second- and third-order Hermitian–Toeplitz determinants for the classes of starlike functions associated with the modified sigmoid function and a related function, whose Taylor coefficients are the Bell numbers, are investigated. Further, the third and fourth Hankel determinants for these classes are also...

The class of close-to-convex functions are univalent and so its subclasses. For normalised analytic functions defined on the unit disk, four subclasses of close-to-convex functions are considered and Hermitian-Toeplitz determinants for these classes are investigated. All the results presented in this article are sharp.

The present work is an attempt to give partial proofs of certain conjectures on the fifth coefficient of certain normalized analytic functions. Further, bounds on the sixth and seventh coefficients for the starlike functions related to a lune are also investigated. The non-sharp bound on third and fourth Hankel determinants are also obtained.

The sharp upper and lower bounds for the third-order Hermitian-Toeplitz determin-
ant are investigated for the classes of Janowski type starlike and convex functions. The results presented in this paper generalize several recent works in this direction.

The famous Littlewood-Paley conjecture is true for the starlike functions but it does not hold for close-to-convex functions. In fact this conjecture does not hold for many well-defined subclasses of normalized univalent functions. The present work considers the classes of strongly $\alpha$-logarithmic close-to-convex and logarithmic $\alpha$-quasi...

There exists a rich literature on the Hankel determinants in the field of geometric function theory. Particularly, it is not easy to find out the sharp bound on the third Hankel determinant as compared to calculate the sharp bound on the second Hankel determinant. The present paper is an attempt to improve certain existing bound on the third Hankel...

Sharp arc length of the image curve of a given line segment joining the points $r^{it}$ and $-re^{it}\;(0<r<1,\;0\leq t\leq2\pi)$ under the Janowski starlike and convex functions are derived. Further, length of image curve of the circle $|z|=r<1$ under the Janowski convex functions is also obtained. Several other results related to the arc length f...

In this manuscript, a conjecture related to the estimate on the fifth coefficient of Bazilevič functions is settled for the range \(1\le \alpha \le \alpha ^*(\approx 2.049)\). However, for \(\alpha >\alpha ^*\), a non-sharp bound on the same is also derived. At the end of this manuscript, sharp upper bound on the functional \(|a_2a_3-a_4|\) is also...

Abstract The conjecture proposed by Raina and Sokòł [Hacet. J. Math. Stat. 44(6):1427–1433 (2015)] for a sharp upper bound on the fourth coefficient has been settled in this manuscript. An example is constructed to show that their conjectures for the bound on the fifth coefficient and the bound related to the second Hankel determinant are false. Ho...

Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several...

The subclasses of univalent functions are closely related to the class of functions with positive real part, also known as the class of Carath\'eodory functions. Several information about the subclasses of univalent functions can be inferred, just by associating them with a suitable function in the Carath\'eodory class. The coefficient problems are...

The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with the Bell numbers are determined.

Abstract. The concept of convolution is applied to investigate some subordination results for the normalized analytic functions whose first derivative belongs to the class of the tilted Caratheodory functions. The sharp radius of starlikeness of order $\alpha$ of the product of two normalized analytic functions satisfying certain specified conditio...

The main aim of this manuscript is to investigate sharp bound on the functional | ap+1ap+2- ap+3| for functions f(z) = zp+ ap+1zp⁺¹+ ap+2zp⁺²+ ap+3zp⁺³+ ⋯ belonging to the class Rp(α) associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on | ap+1ap+2- ap+3| , and |ap+1ap+3-ap+22| for functions in the class RLp...

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further,...

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecuti...

For given real numbers $\alpha$ and $\beta\;(\alpha < 1 <\beta)$, let ${\mathcal P}(\alpha, \beta)$ be the class of analytic functions $p$ with $p(0)=1$ satisfying $\alpha < \RE \{ p(z) \}< \beta$ in the open unit disk $\mathbb{D}:= \set{ z\in\mathbb{C}:|z|<1 }$. For $|z|=r<1$, the lower and upper bounds on the real and imaginary parts for the anal...

We derive sharp upper bound on the initial coefficients and Hankel determinants for normalized analytic functions belonging to a class, introduced by Silverman, defined in terms of ratio of analytic representation of convex and starlike functions. A conjecture related to the coefficients for functions in this class is posed and verified for the fir...

Let $\mathcal{S}^*_s$ be the class of normalized analytic functions $f$ defined on the unit disk such that the quantity $zf'(z)/f(z)$ lies in an eight-shaped region in the right-half plane, which is the image of the unit disk under an entire function defined by $\varphi(z)=1+\sin z$. For this class, we determine the $\mathcal{S}^*_s$-radii for the...

The class $\bar{\mathcal{B}}(\alpha)$ of non-Bazilevi\v c functions was introduced by Obradovi\'c. Later, estimates on the second coefficient and Fekete--Szeg\"{o} functional for normalized analytic functions in the class $\bar{\mathcal{B}}(\alpha)$ were investigated by Tuneski and Darus. In the present work, sharp estimate on third to eighth coeff...

We obtain several inclusions between the class of functions with positive real part and the class of starlike univalent functions associated with the Booth lemniscate. These results are proved by applying the well-known theory of differential subordination developed by Miller and Mocanu and these inclusions give sufficient conditions for normalized...

The main aim of this manuscript is to investigate sharp bound on the functional $|a_{p+1}a_{p+2}-a_{p+3}|$ for functions $f(z)=z^p+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+a_{p+3}z^{p+3}+\cdots$ belonging to the class $\mathcal{R}_p(\alpha)$ associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on $|a_{p+1}a_{p+2}-a_{p+3}...

Sharp radius constants for certain classes of normalized analytic functions
with ﬁxed second coefficient, to be in the classes of starlike functions of positive order,
parabolic starlike functions, and Sokol-Stankiewicz starlike functions are obtained. Our
results extend several earlier works.

Inspired by the work of Srivastava and Patel [Applications of differential subordination to certain subclasses of meromorphically multivalent functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 3, Article 88, 15 pp.], in the present manuscript, using the principle of differential subordination, certain interesting results such as subordination pr...

In the present investigation, we introduce $\mathcal{S}^{\alpha}_g(\phi),$ the class of functions $f\in\mathcal{A}$ satisfying $$1+\frac{z(f*g)'(z))}{(f*g)(z)}+\frac{z(f*g)''(z)}
{(f*g)'(z)}-\frac{(1-\alpha) z^2(f*g)''(z)+z(f*g)'(z)}
{(1-\alpha) z(f*g)'(z)+\alpha(f*g)(z)}\prec\phi(z)\;\; (\alpha\geq0, z\in\mathbb{U})$$ where $g$ is a fixed normaliz...

A generalized linear operator Og,h(α) is deﬁned on the space of normalized analytic
functions for each pair (g, h) of normalized analytic functions. In the present investiga-
tion diﬀerential subordination, diﬀerential superordination and corresponding sandwich
results are obtained for this generalized linear operator Og,h(α) as well as some releva...

In the present investigation, we derive Fekete-Szeg\"{o} inequality for the
class $\mathcal{S}^{\alpha}_{\mathscr{L}_{g}}(\phi)$, introduced here. In
addition to that, certain applications of our results are also discussed.

In this paper, certain linear operators defined on $p$-valent analytic
functions have been unified and for them some subordination and superordination
results as well as the corresponding sandwich type results are obtained. A
related integral transform is discussed and sufficient conditions for functions
in different classes have been obtained.

Let $g$ and $h$ be two fixed normalized analytic functions and $\phi$ be starlike with respect to $1$ whose range is symmetric with respect to the real axis. Let $\mathcal{M}^{\alpha,\beta}_{g,h}(\phi),$ be the class of analytic functions $f(z)=z+a_2z^2+a_3z^3+\ldots$, satisfying
$$\left(\frac{(f*g)(z)}{z}\right)^\alpha \left(\frac{(f*h)(z)}{z}\rig...

Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta
zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}.
Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or
p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies
p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that conditi...

In the present investigation, by taking φ(z) as an analytic function, sharp upper bounds of the Fekete-Szegö functional |a3 − µa 2 2 | for functions be-longing to the class M α g,h (φ) are obtained. A few applications of our main result are also discussed.

A bi-univalent function is a univalent function defined on the unit disk with
its inverse also univalent on the unit disk. Estimates for the initial
coefficients are obtained for bi-univalent functions belonging to certain
classes defined by subordination and relevant connections with earlier results
are pointed out.