ABSTRACT Some new results concerning Simpson type numerical integration are proved, discussed and... more ABSTRACT Some new results concerning Simpson type numerical integration are proved, discussed and compared with other similar results in the literature. A technique to improve the error estimates also in some other recent related results is pointed out.
In this paper we prove some inequalities for convex function of a higher order. The Abel - Gontsc... more In this paper we prove some inequalities for convex function of a higher order. The Abel - Gontscharoff interpolating polynomial with two- point right focal conditions leads us to an inequality like converse of Jensen inequality for a regular, signed measure and, as a consequence, some generalization of Hadamard and Petrovic's inequalities.
"In this paper we give some results which are connected to the parallelogram law and valid i... more "In this paper we give some results which are connected to the parallelogram law and valid in $n$-inner product spaces. We also prove some related results for Bohr's inequality in $n$-inner product spaces."
In this paper we consider the Cauchy's error representation of Lidstone interpolating polynomial ... more In this paper we consider the Cauchy's error representation of Lidstone interpolating polynomial and as a consequence the results concerning to the Hermite-Hadamard inequalities. Using these inequalities, we produce new exponentially convex functions. Also, we give several examples of the families of functions for which the obtained results can be applied.
Kesava Menon's classical inequality is connected with the tails of Taylor's series. In this paper... more Kesava Menon's classical inequality is connected with the tails of Taylor's series. In this paper the original statements are improved, under the same or weaker conditions on functions involved.
ABSTRACT A new generalization and unification of some recent generalizations of Simpson, trapezoi... more ABSTRACT A new generalization and unification of some recent generalizations of Simpson, trapezoid and Ostrowski type inequalities for convex functions is proved and discussed. Both the case with usual canonical partition and generalized parti-tion are considered.
Some new inequalities (error estimates) concerning weighted Simp- son type numerical integration ... more Some new inequalities (error estimates) concerning weighted Simp- son type numerical integration for suitable function spaces and norms are proved, discussed and compared with similar results in the literature. As a natural preparation we also prove an extension of a weighted form of the Montgomery identity. As applications some new estimates for the expectation of a random variable X or, more generally a function of X, e.g. the r− moment Xr, are pointed out.
Mathematical Inequalities & Applications, 2013
In this paper we consider (2n) -convex functions and completely convex functions. Using Lidstone'... more In this paper we consider (2n) -convex functions and completely convex functions. Using Lidstone's interpolating polynomials and conditions on Green's functions we present results for Jensen's inequality and converses of Jensen's inequality for signed measure. By using the obtained inequalities, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied.
In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generaliza... more In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrović's inequality.
ABSTRACT A sequence of inequalities which include McShane’s generalization of Jensen’s inequality... more ABSTRACT A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.
Mathematical Inequalities & Applications, 2012
In this paper we obtain a formula which involves the n th derivative of the first order divided d... more In this paper we obtain a formula which involves the n th derivative of the first order divided difference and a corresponding inequality for functions whose (n + 1)-th derivative belongs to a L p space. These results are a generalization of the results from [1] and [2]. Finally, some examples are given.
Using appropriate functions in Hayashi's inequality we shall establish three extensions of l... more Using appropriate functions in Hayashi's inequality we shall establish three extensions of lyengar's inequality. We shall also improve and generalize an inequality associated with Lidstone polynomials.
The object is to give an overview of the study of Schur-convexity of various means and functions ... more The object is to give an overview of the study of Schur-convexity of various means and functions and to contribute to the subject with some new results. First, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established, and some applications of the extended result are given. Furthermore, Schur-convexity of functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered. 1 If g is convex (concave) and Φ is increasing and Schur-convex (Schur-concave), then Ψ is Schur-convex (Schur-concave). 2 If g is concave (convex) and Φ is decreasing and Schur-convex (Schur-concave), then Ψ is Schur-convex (Schur-concave).
ABSTRACT We give some results which are in connection to the parallelogram law in G-inner product... more ABSTRACT We give some results which are in connection to the parallelogram law in G-inner product spaces and also prove some results related to Bohr’s inequality in G-inner product spaces.
ABSTRACT We shall present several generalizations of discrete Wirtinger's inequality, and... more ABSTRACT We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.
ABSTRACT Some new results concerning Simpson type numerical integration are proved, discussed and... more ABSTRACT Some new results concerning Simpson type numerical integration are proved, discussed and compared with other similar results in the literature. A technique to improve the error estimates also in some other recent related results is pointed out.
In this paper we prove some inequalities for convex function of a higher order. The Abel - Gontsc... more In this paper we prove some inequalities for convex function of a higher order. The Abel - Gontscharoff interpolating polynomial with two- point right focal conditions leads us to an inequality like converse of Jensen inequality for a regular, signed measure and, as a consequence, some generalization of Hadamard and Petrovic's inequalities.
"In this paper we give some results which are connected to the parallelogram law and valid i... more "In this paper we give some results which are connected to the parallelogram law and valid in $n$-inner product spaces. We also prove some related results for Bohr's inequality in $n$-inner product spaces."
In this paper we consider the Cauchy's error representation of Lidstone interpolating polynomial ... more In this paper we consider the Cauchy's error representation of Lidstone interpolating polynomial and as a consequence the results concerning to the Hermite-Hadamard inequalities. Using these inequalities, we produce new exponentially convex functions. Also, we give several examples of the families of functions for which the obtained results can be applied.
Kesava Menon's classical inequality is connected with the tails of Taylor's series. In this paper... more Kesava Menon's classical inequality is connected with the tails of Taylor's series. In this paper the original statements are improved, under the same or weaker conditions on functions involved.
ABSTRACT A new generalization and unification of some recent generalizations of Simpson, trapezoi... more ABSTRACT A new generalization and unification of some recent generalizations of Simpson, trapezoid and Ostrowski type inequalities for convex functions is proved and discussed. Both the case with usual canonical partition and generalized parti-tion are considered.
Some new inequalities (error estimates) concerning weighted Simp- son type numerical integration ... more Some new inequalities (error estimates) concerning weighted Simp- son type numerical integration for suitable function spaces and norms are proved, discussed and compared with similar results in the literature. As a natural preparation we also prove an extension of a weighted form of the Montgomery identity. As applications some new estimates for the expectation of a random variable X or, more generally a function of X, e.g. the r− moment Xr, are pointed out.
Mathematical Inequalities & Applications, 2013
In this paper we consider (2n) -convex functions and completely convex functions. Using Lidstone'... more In this paper we consider (2n) -convex functions and completely convex functions. Using Lidstone's interpolating polynomials and conditions on Green's functions we present results for Jensen's inequality and converses of Jensen's inequality for signed measure. By using the obtained inequalities, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied.
In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generaliza... more In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrović's inequality.
ABSTRACT A sequence of inequalities which include McShane’s generalization of Jensen’s inequality... more ABSTRACT A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.
Mathematical Inequalities & Applications, 2012
In this paper we obtain a formula which involves the n th derivative of the first order divided d... more In this paper we obtain a formula which involves the n th derivative of the first order divided difference and a corresponding inequality for functions whose (n + 1)-th derivative belongs to a L p space. These results are a generalization of the results from [1] and [2]. Finally, some examples are given.
Using appropriate functions in Hayashi's inequality we shall establish three extensions of l... more Using appropriate functions in Hayashi's inequality we shall establish three extensions of lyengar's inequality. We shall also improve and generalize an inequality associated with Lidstone polynomials.
The object is to give an overview of the study of Schur-convexity of various means and functions ... more The object is to give an overview of the study of Schur-convexity of various means and functions and to contribute to the subject with some new results. First, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established, and some applications of the extended result are given. Furthermore, Schur-convexity of functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered. 1 If g is convex (concave) and Φ is increasing and Schur-convex (Schur-concave), then Ψ is Schur-convex (Schur-concave). 2 If g is concave (convex) and Φ is decreasing and Schur-convex (Schur-concave), then Ψ is Schur-convex (Schur-concave).
ABSTRACT We give some results which are in connection to the parallelogram law in G-inner product... more ABSTRACT We give some results which are in connection to the parallelogram law in G-inner product spaces and also prove some results related to Bohr’s inequality in G-inner product spaces.
ABSTRACT We shall present several generalizations of discrete Wirtinger's inequality, and... more ABSTRACT We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.
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