Saminathan Ponnusamy

Duality for hadamard products applied to certain integral transforms

Let A1 be the class of functions analytic in the unit disk and normalized by f(0)= f′–1=0. We shall investigate the integral transform where λ is a non-negative real valued function normalized by From our main results we get conditions on the number β and the function λ such that V λ(f) is starlike of order γ(0 ≤ γ ≤ 1/2) when f has the property that Re[e iα (f′(z)–β)]> 0. We also find conditions implying that F=V λ(f) satisfies the starlikeness condition . As examples we study various choices of λ(t), related to classical integral transforms.

Starlikeness and convexity of generalized Bessel functions

In this paper, we give sufficient conditions for the parameters of the normalized form of the generalized Bessel functions to be convex and starlike in the open unit disk. As an application of our main results, we solve a recent open problem concerning a subordination property of Bessel functions with different parameters. Moreover, we present a new inequality for the Euler gamma function, which we apply in order to have tight bounds for the generalized and normalized Bessel function of the first kind.

New criteria and distortion theorems for univalent functions

Let denote the family of functions f, normalized by f(0)=0=f′−1, that are analytic in the open unit disc and such that for some with . The main object of this paper is to study this class and to find conditions on α, β and on the function g such that each function f in belongs to a family which is contained in the family of univalent functions in the unit disc δ. We also find the exact value of in the class for fixed zεδ. Further, we also determine condition on λ for functions f in P(λ) to be in the class of strongly starlike functions, or in the class of functions whose derivative lies in a sector of angle less than or equal to πγ/2 with γε(0,1]. Finally, we also obtain a sufficient condition for an analytic function f to satisfy the analytic univalence criteria of Noshiro-Warschawski. Several examples are stated in support of the sharpness of our criteria.

On Turán type inequalities for modified Bessel functions

In this note our aim is to point out that certain inequalities for modified Bessel func- tions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equiva- lent to the corresponding Turan type inequalities for these functions. Moreover, we present some new Turan type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture is posed, which may be of interest for further research.

Close-to-convexity properties of Gaussian hypergeometric functions

Abstract Let A be the class of normalized analytic functions in the unit disk Δ. Let φ ( z ) be either zF ( a , b ; c ; z ) or ( c ab )[F(a,b;c;z) − 1] , where F ( a , b ; c ; z ) denotes the classical hypergeometric function. The purpose of this paper is to study close-to-convexity (and hence univalency) of φ ( z ) in the unit disc. More generally, we find conditions on a , b , c and β such that φ satisfies Re e iη ((1 − z ) φ ′( z ) − β ) > 0 for all z ∈ Δ and for some real η ∈ ( −1 2 π, 1 2 π) .

Criteria for strongly starlike functions

Let A n be the family of normalized regular functions f, where f(z)=z+an+1zn+1+…, in the unit disc Δ and let S(α) and T λ be the families of functions f such that respectively. S(α) is the class of strongly starlike functions. Further, let S ∗(α) denote the well-known family of starlike functions of order α. The purpose of this paper is to find conditions so that 1. satisfying satisfying . 2. satisfying is in S(α) or in 3. satisfying is in S(α) or in S ∗(α); 4. satisfying Re f′(z)≥1−λ1,|g′(z)−1|≤λ2, imply z(f∗g)′ is in S(α). The results of this paper extend the previously known results and improve some criteria for starlike functions. These are then used to obtain new information for uniformly convex functions and Bemardi integral transform.

A class of locally univalent functions defined by a differential inequality

We study the range of parameters λ and μ such that any function f (z), analytic for |z| < 1 with is starlike or spirallike. §Dedicated to the memory of Vikramaditya Singh.

Coefficient conditions for harmonic univalent mappings and hypergeometric mappings

In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (resp. fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provide examples of harmonic polynomials that are close-to-convex and starlike, respectively.

Landau’s theorem for certain biharmonic mappings

Abstract In this paper, we show the existence of Landau and Bloch constants for biharmonic mappings of the form L ( F ) . Here L represents the linear complex operator L = z ∂ ∂ z - z ¯ ∂ ∂ z ¯ defined on the class of complex-valued C 1 functions in the plane, and F belongs to the class of biharmonic mappings of the form F ( z ) = | z | 2 G ( z ) + K ( z ) ( | z | 1 ) , where G and K are harmonic.

Hypergeometric transforms of functions with derivative in a half plane

Let A be the class of normalized analytic functions in the unit disk Δ, F(a,b;c;z) and Φ(a;c;z) denote respectively, the Gaussian and confluent hypergeometric functions. Let R(β) = [lcub]ƒ ∈ A: ∃η ∈ R[rcub] such that Re [eiη(ƒ′(z) − β)]>0, z ∈ Δ[rcub]. For ƒ ∈ A, we define the hypergeometric transforms Va,b;c(ƒ) and Ua;c(ƒ) by the convolution Va,b;c(ƒ) := zF(a,b;c;z) ∗ ƒ(z) and Ua;c(ƒ) := zΦ(a;c;z) ∗ ƒ(z), respectively. The main aim of this paper is to find conditions on β1, β2 and the parameters (a,b,c) such that each of the operators Va,b;c(ƒ) and Ua;c(ƒ) maps R(β1) into R(β2). We also find conditions such that the fu (cab)[F(a,b;c;z)−1] or (ca)[Φ(a;c;z) − 1] is in R(β).