Saiful R Mondal | King Faisal University (original) (raw)

Papers by Saiful R Mondal

Symmetry

In this paper, we consider birth and death processes with different sequences of transition rates... more In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.

Mathematics

In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differen... more In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differential equations y″(z)+a(z)y′(z)+b(z)y(z)=0 and z2y″(z)+a(z)zy′(z)+b(z)y(z)=0, such their solution f(z) with normalization f(0)=0=f′(0)−1 is starlike in the lemniscate domain, equivalently zf′(z)/f(z)≺1+z. We provide several examples with graphical presentations for a clear view of the obtained results.

Cornell University - arXiv, Nov 28, 2016

In the present research article, we investigate solutions for fractional kinetic equations, invol... more In the present research article, we investigate solutions for fractional kinetic equations, involving k-Struve functions, some of the salient properties of which we present. The method used is Laplace transform based. the methodology and results can be adopted and extended to various related fractional problems in mathematical physics.

The Korean Mathematical Society, Apr 1, 2019

Open Mathematics, 2015

The main purpose of this paper is to introduce a class of new integrals involving generalized Bes... more The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.

Mathematics

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defin... more In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of differentiable functions. The results are proved using the nth positive zero, denoted by bn(ν). Special cases lead to several examples involving special functions, namely, Bessel, Struve, and Hurwitz functions, as well as several other trigonometric functions.

Symmetry

The function PL(z)=1+z maps the unit disc D={z∈C:|z|<1} to a leminscate which is symmetric abo... more The function PL(z)=1+z maps the unit disc D={z∈C:|z|<1} to a leminscate which is symmetric about the x-axis. The conditions on the parameters α and n, for which the associated Laguerre polynomial (ALP) Lnα maps unit disc into the leminscate domain, are deduced in this article. We also establish the condition under which a function involving Lnα maps D to a domain subordinated by ϕNe(z)=1−z+z3/3, ϕe(z)=ez, and ϕA(z)=1+Az, A∈(0,1]. We provide several graphical presentations for a clear view of some of the obtained results. The possibilities for the improvements of the results are also highlighted.

Communications of the Korean Mathematical Society, 2017

Generalized integral formulas involving the generalized modified k-Bessel function J b,c,γ,λ k,ν ... more Generalized integral formulas involving the generalized modified k-Bessel function J b,c,γ,λ k,ν (z) of first kind are expressed in terms generalized Wright functions. Some interesting special cases of the main results are also discussed

Axioms

The primary objective of this study is to establish necessary conditions so that the normalized F... more The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k-uniform convexity and k-uniform starlikeness. Furthermore, some sufficient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples.

Symmetry

The functions 1+z, ez, 1+Az, A∈(0,1] map the unit disc D to a domain which is symmetric about the... more The functions 1+z, ez, 1+Az, A∈(0,1] map the unit disc D to a domain which is symmetric about the x-axis. The Regular Coulomb wave function (RCWF) FL,η is a function involving two parameters L and η, and FL,η is symmetric about these. In this article, we derive conditions on the parameter L and η for which the normalized form fL of FL,η are subordinated by 1+z. We also consider the subordination by ez and 1+Az, A∈(0,1]. A few more subordination properties involving RCWF are discussed, which leads to the star-likeness of normalized Regular Coulomb wave functions.

Symmetry

This article considers three types of analytic functions based on their infinite product represen... more This article considers three types of analytic functions based on their infinite product representation. The radius of the k-parabolic starlikeness of the functions of these classes is studied. The optimal parameter values for k-parabolic starlike functions are determined in the unit disk. Several examples are provided that include special functions such as Bessel, Struve, Lommel, and q-Bessel functions.

Sufficient conditions on A, B, p, b and c are determined that will ensure the generalized Bessel ... more Sufficient conditions on A, B, p, b and c are determined that will ensure the generalized Bessel functions u_p,b,c satisfies the subordination u_p,b,c(z) ≺ (1+Az)/ (1+Bz). In particular this gives conditions for (-4κ/c)(u_p,b,c(z)-1), c ≠ 0 to be close-to-convex. Also, conditions for which u_p,b,c(z) to be Janowski convex, and zu_p,b,c(z) to be Janowski starlike in the unit disk D={z ∈C: |z|<1} are obtained.

For a fixed a ∈{1, 2, 3, ...}, the radius of starlikeness of positive order is obtained for each ... more For a fixed a ∈{1, 2, 3, ...}, the radius of starlikeness of positive order is obtained for each of the normalized analytic functions f_a, ν(z)&:= (2^a ν-a+1 a^-a(aν-a+1)/2Γ(a ν+1) _aB_2a-1, a ν-a+1, 1(a^a/2 z))^1a ν-a+1, g_a, ν(z)&:= 2^a ν-a+1 a^-a/2(aν-a+1)Γ(a ν+1) z^a-aν_aB_2a-1, a ν-a+1, 1(a^a/2 z), h_a, ν(z)&:= 2^a ν-a+1 a^-a/2(aν-a+1)Γ(a ν+1) z^1/2(1+a-aν)_aB_2a-1, a ν-a+1, 1(a^a/2√(z)) in the unit disk, where _aB_b, p, c is the generalized Bessel function _aB_b, p, c(z):= ∑_k=0^∞(-c)^k/k! Γ( a k +p+b+1/2)(z/2)^2k+p. The best range on ν is also obtained for a fixed a to ensure the functions f_a, ν and g_a, ν are starlike of positive order in the unit disk. When a=1, the results obtained reduced to earlier known results.

arXiv: Classical Analysis and ODEs, 2013

Two integral operator involving the Appell's functions, or Horn's function in the kernel ... more Two integral operator involving the Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind are expressed in term of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function are also discussed.

International Journal of Analysis, 2016

Sufficient conditions on A, B, p, b, and c are determined that will ensure the generalized Bessel... more Sufficient conditions on A, B, p, b, and c are determined that will ensure the generalized Bessel function up,b,c satisfies the subordination up,b,c(z)≺1+Az/(1+Bz). In particular this gives conditions for (-4κ/c)(up,b,c(z)-1), c≠0, to be close-to-convex. Also, conditions for up,b,c(z) to be Janowski convex and zup,b,c(z) to be Janowski starlike in the unit disk D=z∈C:z<1 are obtained.

Bulletin of the Korean Mathematical Society, 2016

In this paper our aim is to study the classical Bessel-Struve kernel. Monotonicity and log-convex... more In this paper our aim is to study the classical Bessel-Struve kernel. Monotonicity and log-convexity properties for the Bessel-Struve kernel, and the ratio of the Bessel-Struve kernel and the Kummer confluent hypergeometric function are investigated. Moreover, lower and upper bounds are given for the Bessel-Struve kernel in terms of the exponential function and some Turán type inequalities are deduced.

Results in Mathematics, 2011

In this work, using subordination results for the Cesáro sum of certain analytic functions, a con... more In this work, using subordination results for the Cesáro sum of certain analytic functions, a concept called Cesáro stable is given and two conjectures in this direction are also proposed. Using an extension of Vietoris' theorem on positivity of cosine sums, some results on positivity of cosine and sine sums are obtained. These results are useful in discussing some particular cases of the proposed conjectures.

Mathematical Problems in Engineering, 2014

Two integral operators involving Appell's functions, or Horn's function in the kernel are... more Two integral operators involving Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind is expressed in terms of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function, are also discussed.

Journal of Inequalities and Applications, 2013

For a prestarlike function f of nonnegative order α, 0 ≤ α < 1, and a close-to-convex function zg... more For a prestarlike function f of nonnegative order α, 0 ≤ α < 1, and a close-to-convex function zg of order α, the convolution g * f is shown to be zero-free in the open unit disk. The result can be applied to a wide spectrum of interesting approximants, including those involving the Cesàro means and Jacobi polynomials. If zg is also prestarlike, then the range of g * f is shown to be contained in a sector with opening angle strictly less than 2π .

Symmetry

In this paper, we consider birth and death processes with different sequences of transition rates... more In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.

Mathematics

In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differen... more In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differential equations y″(z)+a(z)y′(z)+b(z)y(z)=0 and z2y″(z)+a(z)zy′(z)+b(z)y(z)=0, such their solution f(z) with normalization f(0)=0=f′(0)−1 is starlike in the lemniscate domain, equivalently zf′(z)/f(z)≺1+z. We provide several examples with graphical presentations for a clear view of the obtained results.

Cornell University - arXiv, Nov 28, 2016

In the present research article, we investigate solutions for fractional kinetic equations, invol... more In the present research article, we investigate solutions for fractional kinetic equations, involving k-Struve functions, some of the salient properties of which we present. The method used is Laplace transform based. the methodology and results can be adopted and extended to various related fractional problems in mathematical physics.

The Korean Mathematical Society, Apr 1, 2019

Open Mathematics, 2015

The main purpose of this paper is to introduce a class of new integrals involving generalized Bes... more The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.

Mathematics

In this paper, we aim to construct inequalities of the Redheffer type for certain functions defin... more In this paper, we aim to construct inequalities of the Redheffer type for certain functions defined by the infinite product involving the zeroes of these functions. The key tools used in our proofs are classical results on the monotonicity of the ratio of differentiable functions. The results are proved using the nth positive zero, denoted by bn(ν). Special cases lead to several examples involving special functions, namely, Bessel, Struve, and Hurwitz functions, as well as several other trigonometric functions.

Symmetry

The function PL(z)=1+z maps the unit disc D={z∈C:|z|<1} to a leminscate which is symmetric abo... more The function PL(z)=1+z maps the unit disc D={z∈C:|z|<1} to a leminscate which is symmetric about the x-axis. The conditions on the parameters α and n, for which the associated Laguerre polynomial (ALP) Lnα maps unit disc into the leminscate domain, are deduced in this article. We also establish the condition under which a function involving Lnα maps D to a domain subordinated by ϕNe(z)=1−z+z3/3, ϕe(z)=ez, and ϕA(z)=1+Az, A∈(0,1]. We provide several graphical presentations for a clear view of some of the obtained results. The possibilities for the improvements of the results are also highlighted.

Communications of the Korean Mathematical Society, 2017

Generalized integral formulas involving the generalized modified k-Bessel function J b,c,γ,λ k,ν ... more Generalized integral formulas involving the generalized modified k-Bessel function J b,c,γ,λ k,ν (z) of first kind are expressed in terms generalized Wright functions. Some interesting special cases of the main results are also discussed

Axioms

The primary objective of this study is to establish necessary conditions so that the normalized F... more The primary objective of this study is to establish necessary conditions so that the normalized Fox–Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox–Wright functions and discuss its k-uniform convexity and k-uniform starlikeness. Furthermore, some sufficient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples.

Symmetry

The functions 1+z, ez, 1+Az, A∈(0,1] map the unit disc D to a domain which is symmetric about the... more The functions 1+z, ez, 1+Az, A∈(0,1] map the unit disc D to a domain which is symmetric about the x-axis. The Regular Coulomb wave function (RCWF) FL,η is a function involving two parameters L and η, and FL,η is symmetric about these. In this article, we derive conditions on the parameter L and η for which the normalized form fL of FL,η are subordinated by 1+z. We also consider the subordination by ez and 1+Az, A∈(0,1]. A few more subordination properties involving RCWF are discussed, which leads to the star-likeness of normalized Regular Coulomb wave functions.

Symmetry

This article considers three types of analytic functions based on their infinite product represen... more This article considers three types of analytic functions based on their infinite product representation. The radius of the k-parabolic starlikeness of the functions of these classes is studied. The optimal parameter values for k-parabolic starlike functions are determined in the unit disk. Several examples are provided that include special functions such as Bessel, Struve, Lommel, and q-Bessel functions.

Sufficient conditions on A, B, p, b and c are determined that will ensure the generalized Bessel ... more Sufficient conditions on A, B, p, b and c are determined that will ensure the generalized Bessel functions u_p,b,c satisfies the subordination u_p,b,c(z) ≺ (1+Az)/ (1+Bz). In particular this gives conditions for (-4κ/c)(u_p,b,c(z)-1), c ≠ 0 to be close-to-convex. Also, conditions for which u_p,b,c(z) to be Janowski convex, and zu_p,b,c(z) to be Janowski starlike in the unit disk D={z ∈C: |z|<1} are obtained.

For a fixed a ∈{1, 2, 3, ...}, the radius of starlikeness of positive order is obtained for each ... more For a fixed a ∈{1, 2, 3, ...}, the radius of starlikeness of positive order is obtained for each of the normalized analytic functions f_a, ν(z)&:= (2^a ν-a+1 a^-a(aν-a+1)/2Γ(a ν+1) _aB_2a-1, a ν-a+1, 1(a^a/2 z))^1a ν-a+1, g_a, ν(z)&:= 2^a ν-a+1 a^-a/2(aν-a+1)Γ(a ν+1) z^a-aν_aB_2a-1, a ν-a+1, 1(a^a/2 z), h_a, ν(z)&:= 2^a ν-a+1 a^-a/2(aν-a+1)Γ(a ν+1) z^1/2(1+a-aν)_aB_2a-1, a ν-a+1, 1(a^a/2√(z)) in the unit disk, where _aB_b, p, c is the generalized Bessel function _aB_b, p, c(z):= ∑_k=0^∞(-c)^k/k! Γ( a k +p+b+1/2)(z/2)^2k+p. The best range on ν is also obtained for a fixed a to ensure the functions f_a, ν and g_a, ν are starlike of positive order in the unit disk. When a=1, the results obtained reduced to earlier known results.

arXiv: Classical Analysis and ODEs, 2013

Two integral operator involving the Appell's functions, or Horn's function in the kernel ... more Two integral operator involving the Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind are expressed in term of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function are also discussed.

International Journal of Analysis, 2016

Sufficient conditions on A, B, p, b, and c are determined that will ensure the generalized Bessel... more Sufficient conditions on A, B, p, b, and c are determined that will ensure the generalized Bessel function up,b,c satisfies the subordination up,b,c(z)≺1+Az/(1+Bz). In particular this gives conditions for (-4κ/c)(up,b,c(z)-1), c≠0, to be close-to-convex. Also, conditions for up,b,c(z) to be Janowski convex and zup,b,c(z) to be Janowski starlike in the unit disk D=z∈C:z<1 are obtained.

Bulletin of the Korean Mathematical Society, 2016

In this paper our aim is to study the classical Bessel-Struve kernel. Monotonicity and log-convex... more In this paper our aim is to study the classical Bessel-Struve kernel. Monotonicity and log-convexity properties for the Bessel-Struve kernel, and the ratio of the Bessel-Struve kernel and the Kummer confluent hypergeometric function are investigated. Moreover, lower and upper bounds are given for the Bessel-Struve kernel in terms of the exponential function and some Turán type inequalities are deduced.

Results in Mathematics, 2011

In this work, using subordination results for the Cesáro sum of certain analytic functions, a con... more In this work, using subordination results for the Cesáro sum of certain analytic functions, a concept called Cesáro stable is given and two conjectures in this direction are also proposed. Using an extension of Vietoris' theorem on positivity of cosine sums, some results on positivity of cosine and sine sums are obtained. These results are useful in discussing some particular cases of the proposed conjectures.

Mathematical Problems in Engineering, 2014

Two integral operators involving Appell's functions, or Horn's function in the kernel are... more Two integral operators involving Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind is expressed in terms of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function, are also discussed.

Journal of Inequalities and Applications, 2013

For a prestarlike function f of nonnegative order α, 0 ≤ α < 1, and a close-to-convex function zg... more For a prestarlike function f of nonnegative order α, 0 ≤ α < 1, and a close-to-convex function zg of order α, the convolution g * f is shown to be zero-free in the open unit disk. The result can be applied to a wide spectrum of interesting approximants, including those involving the Cesàro means and Jacobi polynomials. If zg is also prestarlike, then the range of g * f is shown to be contained in a sector with opening angle strictly less than 2π .