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Papers by Hameed Ur Rehman

Research paper thumbnail of Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

Malaysian Journal of Mathematical Sciences, 2024

The article aims to estimate the coefficient bounds for the second Hankel determinant by using th... more The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained.

Research paper thumbnail of Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Article, 2021

In the present paper, the authors implement the two analytic functions with its positive real par... more In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. is formula is applied to a certain class of bi-univalent functions and solve the n-th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n-th term.

Research paper thumbnail of The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

Article, 2022

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(... more The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) \u003e 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Research paper thumbnail of The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

Mathematics and Statistics, 2022

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(... more The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Research paper thumbnail of Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Journal of Mathematics, 2021

In the present paper, the authors implement the two analytic functions with its positive real par... more In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. is formula is applied to a certain class of bi-univalent functions and solve the n-th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n-th term.

Research paper thumbnail of Normalization of the generalized K− Mittag-Leffler function and ratio to its sequence of partial sums

Article, 2018

In this article we introduce an operator L γ,q k,α (β, δ)(f)(z) associated with generalized K− Mi... more In this article we introduce an operator L γ,q k,α (β, δ)(f)(z) associated with generalized K− Mittag-Leffler function in the unit disk U= {z : |z| < 1}. Further the ratio of normalized K− Mittag-Leffer function Q γ,q k,α,β,δ (z) to its sequence of partial sums Q(γ,q k,α,β,δ) m (z) are calculated. M.S.C. 2010: 30C45.

Research paper thumbnail of Graphing Examples of Starlike and Convex Functions of order β

Articl, 2018

In this paper, we introduce and investigate two subclasses H * β (n, b, φ) and K β (n, b, φ) of a... more In this paper, we introduce and investigate two subclasses H * β (n, b, φ) and K β (n, b, φ) of analytic functions with negative coefficients of order β in the unit disk D. Silverman [6] determined certain coefficient inequalities and distortion theorems for univalent functions with negative coefficients that are starlike S(α), and convex K(α) of order α respectively. We also obtain the same coefficient inequalities and distortion theorem for such univalent functions with negative coefficients. We point out that the coefficient estimates up to z 9 are enough to get the graphs of starlike and convex functions. Here in the sequel we give some examples, estimation and the graphical representation of such univalent functions by using the complex tool [8].

Research paper thumbnail of COEFFICIENT PROPERTIES INVOLVING THE GENERALIZED K−MITTAG-LEFFLER FUNCTIONS

Article, 2017

In this article we investigate the Fekete-Szegö problem for the integral operator associated with... more In this article we investigate the Fekete-Szegö problem for the integral operator associated with the most generalized K− Mittag-Leffler function. Our results will focus on some of the subclasses of starlike and convex functions.

Research paper thumbnail of A Note on Caputo Derivative Operator interpretation in Economy

Article, 2018

We propound the economic idea in terms of fractional derivatives, which involves the modified Cap... more We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo's fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of T- Indicators which analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo's derivative operator of order 1/2 instead of integer order derivative í µí±›. The two sides of Caputo's derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo's operator.

Research paper thumbnail of Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

Malaysian Journal of Mathematical Sciences, 2024

The article aims to estimate the coefficient bounds for the second Hankel determinant by using th... more The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained.

Research paper thumbnail of Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Article, 2021

In the present paper, the authors implement the two analytic functions with its positive real par... more In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. is formula is applied to a certain class of bi-univalent functions and solve the n-th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n-th term.

Research paper thumbnail of The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

Article, 2022

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(... more The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) \u003e 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Research paper thumbnail of The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

Mathematics and Statistics, 2022

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(... more The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Research paper thumbnail of Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Journal of Mathematics, 2021

In the present paper, the authors implement the two analytic functions with its positive real par... more In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. is formula is applied to a certain class of bi-univalent functions and solve the n-th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n-th term.

Research paper thumbnail of Normalization of the generalized K− Mittag-Leffler function and ratio to its sequence of partial sums

Article, 2018

In this article we introduce an operator L γ,q k,α (β, δ)(f)(z) associated with generalized K− Mi... more In this article we introduce an operator L γ,q k,α (β, δ)(f)(z) associated with generalized K− Mittag-Leffler function in the unit disk U= {z : |z| < 1}. Further the ratio of normalized K− Mittag-Leffer function Q γ,q k,α,β,δ (z) to its sequence of partial sums Q(γ,q k,α,β,δ) m (z) are calculated. M.S.C. 2010: 30C45.

Research paper thumbnail of Graphing Examples of Starlike and Convex Functions of order β

Articl, 2018

In this paper, we introduce and investigate two subclasses H * β (n, b, φ) and K β (n, b, φ) of a... more In this paper, we introduce and investigate two subclasses H * β (n, b, φ) and K β (n, b, φ) of analytic functions with negative coefficients of order β in the unit disk D. Silverman [6] determined certain coefficient inequalities and distortion theorems for univalent functions with negative coefficients that are starlike S(α), and convex K(α) of order α respectively. We also obtain the same coefficient inequalities and distortion theorem for such univalent functions with negative coefficients. We point out that the coefficient estimates up to z 9 are enough to get the graphs of starlike and convex functions. Here in the sequel we give some examples, estimation and the graphical representation of such univalent functions by using the complex tool [8].

Research paper thumbnail of COEFFICIENT PROPERTIES INVOLVING THE GENERALIZED K−MITTAG-LEFFLER FUNCTIONS

Article, 2017

In this article we investigate the Fekete-Szegö problem for the integral operator associated with... more In this article we investigate the Fekete-Szegö problem for the integral operator associated with the most generalized K− Mittag-Leffler function. Our results will focus on some of the subclasses of starlike and convex functions.

Research paper thumbnail of A Note on Caputo Derivative Operator interpretation in Economy

Article, 2018

We propound the economic idea in terms of fractional derivatives, which involves the modified Cap... more We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo's fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of T- Indicators which analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo's derivative operator of order 1/2 instead of integer order derivative í µí±›. The two sides of Caputo's derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo's operator.