Ravichandran Vaithiyanathan | University of Delhi (original) (raw)

Uploads

Papers by Ravichandran Vaithiyanathan

The authors derive several inequalities associated with differential subordinations between analy... more The authors derive several inequalities associated with differential subordinations between analytic functions and a linear operator defined for a certain family of p-valent functions, which is introduced here by means of this linear operator. Some special cases and consequences of the main results are also considered.

Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.

In this present investigation, we obtain some results for certain second order linear differentia... more In this present investigation, we obtain some results for certain second order linear differential subordination. We also discuss some applications of our results.

In this paper, we determine conditions on β, α i and f i (z) so that the integral operator β

For 0leqalpha<10\leq \alpha <10leqalpha<1, the sharp radii of starlikeness and convexity of order alpha\alphaalpha for functi... more For 0leqalpha<10\leq \alpha <10leqalpha<1, the sharp radii of starlikeness and convexity of order alpha\alphaalpha for functions of the form f(z)=z+a2z2+a3z3+...f(z)=z+a_2z^2+a_3z^3+...f(z)=z+a2z2+a3z3+... whose Taylor coefficients ana_nan satisfy the conditions ∣a2∣=2b|a_2|=2b∣a2∣=2b, 0leqbleq10\leq b\leq 10leqbleq1, and ∣an∣leqn|a_n|\leq n ∣an∣leqn, MMM or M/nM/nM/n ($M>0$) for ngeq3n\geq 3ngeq3 are obtained. Also a class of functions related to Carath\'eodory functions is considered.

By using the theory of first-order differential subordination for functions with fixed initial co... more By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.

A normalized univalent function is uniformly convex if it maps every circular arc contained in th... more A normalized univalent function is uniformly convex if it maps every circular arc contained in the open unit disk with center in it into a convex curve. This article surveys recent results on the class of uniformly convex functions and on an analogous class of uniformly starlike functions.

For certain meromorphic function g and h, we study a class of functions f (z) = z −1 +

Mathematical and Computer Modelling, 2011

Two integral operators on the classes consisting of normalized pp-valent Ma–Minda type starlike a... more Two integral operators on the classes consisting of normalized pp-valent Ma–Minda type starlike and convex functions are considered. Functions in these classes have the form zf′(z)/f(z)≺pφ(z)zf′(z)/f(z)≺pφ(z) and 1+zf″(z)/f′(z)≺pφ(z)1+zf″(z)/f′(z)≺pφ(z) respectively, where φφ is a convex function with φ(0)=1φ(0)=1. It is shown that the first of these operators maps starlike functions into convex functions, while the convex mappings are shown to be closed under the second integral operator.

In this paper, we obtain a subordination result for a class of meromorphic functions.

In this paper, certain linear operators defined on ppp-valent analytic functions have been unifie... more In this paper, certain linear operators defined on ppp-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.

In the present investigation, certain subclasses of close-to-convex functions are investigated. I... more In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szeg\"o functional for functions belonging to the class, distortion, growth estimates and covering theorems.

A normalized analytic function f defined on the open unit disk in the complex plane is in the cla... more A normalized analytic function f defined on the open unit disk in the complex plane is in the class SL if zf'(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w^2 - 1| < 1. In the present investigation, the SL-radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes

Applied Mathematics Letters, 2011

A normalized analytic function ff defined on the open unit disk is a Janowski starlike function i... more A normalized analytic function ff defined on the open unit disk is a Janowski starlike function if zf′(z)/f(z)zf′(z)/f(z) is subordinated to (1+Az)/(1+Bz)(1+Az)/(1+Bz), where AA and BB are complex numbers satisfying the conditions |B|≤1|B|≤1 and A≠BA≠B. In this paper, a new class of analytic functions defined by means of subordination is introduced. Sufficient conditions are obtained for functions in this class to be Janowski starlike. The results obtained extend earlier known works.

Bulletin of The Belgian Mathematical Society-simon Stevin, 2009

Subclasses of p-valent starlike and convex functions in the unit disk in the complex plane are in... more Subclasses of p-valent starlike and convex functions in the unit disk in the complex plane are investigated. Every p-valent convex function in a subclass is shown to belong to its corresponding subclass of starlike functions. A necessary and sufficient condition for functions to belong to these classes is obtained. Subordination properties, and sharp distortion, growth, covering and rotation estimates are obtained for these classes. Convolution results with prestarlike functions are also derived.

Let q1q_1q1 and q2q_2q2 belong to a certain class of normalized analytic univalent functions in the o... more Let q1q_1q1 and q2q_2q2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions ppp to satisfy the double subordination chain q1(z)precp(z)precq2(z)q_1(z)\prec p(z)\prec q_2(z)q_1(z)precp(z)precq_2(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Phi\PhiPhi-like functions.

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\i... more For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on beta\betabeta is determined so that fff is either starlike or convex of order alpha\alphaalpha. Several other coefficient inequalities related to certain subclasses are also investigated.

Applied Mathematics and Computation, 2011

A normalized univalent function f is called Ma–Minda starlike or convex if zf′(z)/f(z)≺φ(z)zf′(z)... more A normalized univalent function f is called Ma–Minda starlike or convex if zf′(z)/f(z)≺φ(z)zf′(z)/f(z)≺φ(z) or 1+zf″(z)/f′(z)≺φ(z)1+zf″(z)/f′(z)≺φ(z) where φφ is a convex univalent function with φ(0)=1φ(0)=1. The class of Ma–Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.

Several radii problems are considered for functions f (z) = z + a 2 z 2 + · · · with fixed second... more Several radii problems are considered for functions f (z) = z + a 2 z 2 + · · · with fixed second coeffcient a 2 . For 0 ≤ β < 1, sharp radius of starlikeness of order β for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order β for uniformly convex functions, and sharp radius of strong-starlikeness of order γ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.

The authors derive several inequalities associated with differential subordinations between analy... more The authors derive several inequalities associated with differential subordinations between analytic functions and a linear operator defined for a certain family of p-valent functions, which is introduced here by means of this linear operator. Some special cases and consequences of the main results are also considered.

Radius constants for several classes of analytic functions on the unit disk are obtained. These i... more Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given.

In this present investigation, we obtain some results for certain second order linear differentia... more In this present investigation, we obtain some results for certain second order linear differential subordination. We also discuss some applications of our results.

In this paper, we determine conditions on β, α i and f i (z) so that the integral operator β

For 0leqalpha<10\leq \alpha <10leqalpha<1, the sharp radii of starlikeness and convexity of order alpha\alphaalpha for functi... more For 0leqalpha<10\leq \alpha <10leqalpha<1, the sharp radii of starlikeness and convexity of order alpha\alphaalpha for functions of the form f(z)=z+a2z2+a3z3+...f(z)=z+a_2z^2+a_3z^3+...f(z)=z+a2z2+a3z3+... whose Taylor coefficients ana_nan satisfy the conditions ∣a2∣=2b|a_2|=2b∣a2∣=2b, 0leqbleq10\leq b\leq 10leqbleq1, and ∣an∣leqn|a_n|\leq n ∣an∣leqn, MMM or M/nM/nM/n ($M>0$) for ngeq3n\geq 3ngeq3 are obtained. Also a class of functions related to Carath\'eodory functions is considered.

By using the theory of first-order differential subordination for functions with fixed initial co... more By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.

A normalized univalent function is uniformly convex if it maps every circular arc contained in th... more A normalized univalent function is uniformly convex if it maps every circular arc contained in the open unit disk with center in it into a convex curve. This article surveys recent results on the class of uniformly convex functions and on an analogous class of uniformly starlike functions.

For certain meromorphic function g and h, we study a class of functions f (z) = z −1 +

Mathematical and Computer Modelling, 2011

Two integral operators on the classes consisting of normalized pp-valent Ma–Minda type starlike a... more Two integral operators on the classes consisting of normalized pp-valent Ma–Minda type starlike and convex functions are considered. Functions in these classes have the form zf′(z)/f(z)≺pφ(z)zf′(z)/f(z)≺pφ(z) and 1+zf″(z)/f′(z)≺pφ(z)1+zf″(z)/f′(z)≺pφ(z) respectively, where φφ is a convex function with φ(0)=1φ(0)=1. It is shown that the first of these operators maps starlike functions into convex functions, while the convex mappings are shown to be closed under the second integral operator.

In this paper, we obtain a subordination result for a class of meromorphic functions.

In this paper, certain linear operators defined on ppp-valent analytic functions have been unifie... more In this paper, certain linear operators defined on ppp-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.

In the present investigation, certain subclasses of close-to-convex functions are investigated. I... more In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szeg\"o functional for functions belonging to the class, distortion, growth estimates and covering theorems.

A normalized analytic function f defined on the open unit disk in the complex plane is in the cla... more A normalized analytic function f defined on the open unit disk in the complex plane is in the class SL if zf'(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w^2 - 1| < 1. In the present investigation, the SL-radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes

Applied Mathematics Letters, 2011

A normalized analytic function ff defined on the open unit disk is a Janowski starlike function i... more A normalized analytic function ff defined on the open unit disk is a Janowski starlike function if zf′(z)/f(z)zf′(z)/f(z) is subordinated to (1+Az)/(1+Bz)(1+Az)/(1+Bz), where AA and BB are complex numbers satisfying the conditions |B|≤1|B|≤1 and A≠BA≠B. In this paper, a new class of analytic functions defined by means of subordination is introduced. Sufficient conditions are obtained for functions in this class to be Janowski starlike. The results obtained extend earlier known works.

Bulletin of The Belgian Mathematical Society-simon Stevin, 2009

Subclasses of p-valent starlike and convex functions in the unit disk in the complex plane are in... more Subclasses of p-valent starlike and convex functions in the unit disk in the complex plane are investigated. Every p-valent convex function in a subclass is shown to belong to its corresponding subclass of starlike functions. A necessary and sufficient condition for functions to belong to these classes is obtained. Subordination properties, and sharp distortion, growth, covering and rotation estimates are obtained for these classes. Convolution results with prestarlike functions are also derived.

Let q1q_1q1 and q2q_2q2 belong to a certain class of normalized analytic univalent functions in the o... more Let q1q_1q1 and q2q_2q2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions ppp to satisfy the double subordination chain q1(z)precp(z)precq2(z)q_1(z)\prec p(z)\prec q_2(z)q_1(z)precp(z)precq_2(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Phi\PhiPhi-like functions.

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\i... more For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on beta\betabeta is determined so that fff is either starlike or convex of order alpha\alphaalpha. Several other coefficient inequalities related to certain subclasses are also investigated.

Applied Mathematics and Computation, 2011

A normalized univalent function f is called Ma–Minda starlike or convex if zf′(z)/f(z)≺φ(z)zf′(z)... more A normalized univalent function f is called Ma–Minda starlike or convex if zf′(z)/f(z)≺φ(z)zf′(z)/f(z)≺φ(z) or 1+zf″(z)/f′(z)≺φ(z)1+zf″(z)/f′(z)≺φ(z) where φφ is a convex univalent function with φ(0)=1φ(0)=1. The class of Ma–Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.

Several radii problems are considered for functions f (z) = z + a 2 z 2 + · · · with fixed second... more Several radii problems are considered for functions f (z) = z + a 2 z 2 + · · · with fixed second coeffcient a 2 . For 0 ≤ β < 1, sharp radius of starlikeness of order β for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order β for uniformly convex functions, and sharp radius of strong-starlikeness of order γ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.