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  1. 20 Demonstratio Mathematica
  2. 14 Journal of Mathematics and Applications
  3. 9 Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka
  4. 5 Annales Polonici Mathematici
  5. 4 Fasciculi Mathematici
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  1. 1 2023
  2. 1 2022
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  1. 6 Dziok J.
  2. 6 Sokół J.
  3. 5 Dragomir S. S.
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1
Content available remote Uniformly convex functions II
100%
Ma W. , Minda D.
|
1993
|
tom 58
|
nr 3
275-285
EN
Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses $f^{-1}(w) = w + d₂w² + d₃w³ + ...$. The series expansion for $f^{-1}(w)$ converges when $|w| < ϱ_f$, where $0 < ϱ_f$ depends on f. The sharp bounds on $|a_n|$ and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on $|a_n|$ and all extremal functions for n = 4, 5, and 6. The same function k and its rotations remain the only extremals. It is known that k and its rotations cannot provide the sharp bound on $|a_n|$ for n sufficiently large. We also find the sharp estimate on the functional |μa²₂ - a₃| for -∞ < μ < ∞. We give sharp bounds on $|d_n|$ for n = 2, 3 and 4. For $n = 2, k^{-1}$ and its rotations are the only extremals. There are different extremal functions for both n = 3 and n = 4. Finally, we show that k and its rotations provide the sharp upper bound on |f''(z)| over the class UCV.
2
Content available remote On regularization in superreflexive Banach spaces by infimal convolution formulas
100%
Cepedello-Boiso M.
Studia Mathematica
|
1998
|
tom 129
|
nr 3
265-284
EN
We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-Hölder derivatives (for some 0 < α≤ 1). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function f which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of Δ-convex $C^{1,α}$ functions converging to f uniformly on bounded sets and preserving the infimum and the set of minimizers of f. The techniques we develop are based on the use of extended inf-convolution formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.
3
Content available remote A generalization of the Hahn-Banach theorem
100%
Plewnia J.
|
1993
|
tom 58
|
nr 1
47-51
EN
If C is a non-empty convex subset of a real linear space E, p: E → ℝ is a sublinear function and f:C → ℝ is concave and such that f ≤ p on C, then there exists a linear function g:E → ℝ such that g ≤ p on E and f ≤ g on C. In this result of Hirano, Komiya and Takahashi we replace the sublinearity of p by convexity.
4
Content available remote Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions
80%
Cerone P. , Dragomir S.
|
2004
|
tom Vol. 37, nr 2
299--308
EN
Ostrowski type inequalities for absolutely continuous functions whose derivatives satisfy certain convexity assumptions are pointed out.
5
Converse Ohlin’s lemma for convex and strongly convex functions
80%
Adamek M. , Nikodem K.
Journal of Applied Analysis
|
2023
|
tom Vol. 29, nr 1
123--126
EN
Theorems which are converse to the Ohlin lemma for convex and strongly convex functions are proved. New proofs of probabilistic characterizations of convex and strongly convex functions are presented.
6
Content available remote Inequalities of Jensen type for φ-convex functions
80%
Dragomir S. S.
Fasciculi Mathematici
|
2015
|
tom Nr 55
35--52
EN
Some inequalities of Jensen type for φ-convex functions defined on real intervals are given.
7
Content available remote Uniformly convex functions
80%
Ma W. , Minda D.
|
1992
|
tom 57
|
nr 2
165-175
EN
Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp bounds on the second and third coefficients. We exhibit a function k in UCV which, up to rotation, is the sole extremal function for these problems. However, we show that this function cannot be extremal for the sharp upper bound on the nth coefficient for all n. We establish this by obtaining the correct order of growth for the sharp upper bound on the nth coefficient over the class UCV and then demonstrating that the nth coefficient of k has a smaller order of growth.
8
Content available remote Euler's Beta function diagonalized and a related functional equation
80%
Choczewski B. , Wach-Michalik A.
Opuscula Mathematica
|
2004
|
tom Vol. 24, no. 1
35-41
EN
Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function beta: R+ → R+, beta(x) := B(x, x), where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if varphi: R+ → R+ satisfies equation (5) and the function h o varphi is continuous and ultimately convex, then varphi = beta.
9
Content available Hermite-Hadamard type inequalities for Wright-convex functions of several variables
80%
Śliwińska D. , Wąsowicz S.
Opuscula Mathematica
|
2015
|
tom Vol. 35, no. 3
411--419
EN
We present Hermite-Hadamard type inequalities for Wright-convex, strongly convex and strongly Wright-convex functions of several variables defined on simplices.
10
Content available remote The Jensen inequality for s-Breckner convex functions in linear spaces
80%
Dragomir S. S. , Fitzpatrick S.
|
2000
|
tom Vol. 33, nr 1
43-49
EN
We derive some inequalities of Jensen's type for 5-convex functions in the sense of Breckner on subsets of linear spaces and give some applications connected with special means.
11
Content available remote On uniformly convex functions
80%
Goodman A.
Annales Polonici Mathematici
|
1991-1992
|
tom 56
|
nr 1
87-92
EN
We introduce a new class of normalized functions regular and univalent in the unit disk. These functions, called uniformly convex functions, are defined by a purely geometric property. We obtain a few theorems about this new class and we point out a number of open problems.
12
Content available remote On a space of entire functions rapidly decreasing on Rn and its Fourier transform
80%
Musin I. d. K.
Concrete Operators
|
2015
|
tom 2
|
nr 1
EN
A space of entire functions of several complex variables rapidly decreasing on Rn and such that their growth along iRn is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on Rn and a Paley-Wiener type theorem are obtained.
13
Content available remote On a subclass of uniformly convex functions with fixed second coefficient
80%
Darwish H.
|
2008
|
tom Vol. 41, nr 4
791-803
EN
Using of Salagean operator, we define a new subclass of uniformly convex functions with negative coefficients and with fixed second coefficient. The main objective of this paper is to obtain coefficient estimates, distortion bounds, closure theorems and extreme points for functions belonging of this new class. The results are generalized to families with fixed finitely many coefficients.
14
Content available remote Hadamard inequalities for Wright-convex functions
70%
Tseng K.-L. , Yang G.-S. , Dragomir S. S.
|
2004
|
tom Vol. 37, nr 3
525--532
EN
In this paper we establish several inequalities of Hadamard's type for Wright-convex functions.
15
Content available remote On the generalization of Wright-convexity
70%
Sobek B.
Journal of Mathematics and Applications
|
2006
|
tom Vol. 28
121-126
16
Content available remote Hermite-Hadamard type inequalities with applications
70%
Khan M. A. , Ali T. , Khan T. U.
Fasciculi Mathematici
|
2017
|
tom Nr 59
57--74
EN
In this article first, we give an integral identity and prove some Hermite-Hadamard type inequalities for the function ƒ such that |ƒ''|q is convex or concave for q ≥ 1. Second, by using these results, we present applications to ƒ-divergence measures. At the end, we obtain some bounds for special means of real numbers and new error estimates for the trapezoidal formula.
17
Content available remote The cancellation law for inf-convolution of convex functions
70%
Zagrodny D.
|
|
nr 3
271-282
EN
Conditions under which the inf-convolution of f and g $f □ g(x):= inf_{y+z=x}(f(y)+g(z))$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions $f: X → ℝ ∪ {+∞}$ on a reflexive Banach space such that $ lim_{∥x∥ → ∞} f(x)/∥x∥ = ∞$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.
18
Content available remote Some inequalities for weighted harmonic and arithmetic operator means
70%
Dragomir S. S.
Fasciculi Mathematici
|
2018
|
tom Nr 61
43--54
EN
In this paper we establish some upper and lower bounds for the difference between the weighted arithmetic and harmonie operator means under various assumption for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.
19
Content available Fekete-Szegö inequalities associated withkthroot transformation based on quasi-subordination
70%
Magesh N. , Yamini J.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2017
|
tom 16
EN
Recently, Haji Mohd and Darus [1] revived the study of coefficient problems for univalent functions associated with quasi-subordination. Inspired largely by this article, we provide coefficient estimates with k-th root transform for certain subclasses of 𝒮 defined by quasi-subordination.
20
Content available remote Monotone Valuations on the Space of Convex Functions
70%
Cavallina L. , Colesanti A.
|
2015
|
tom 3
|
nr 1
EN
We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.
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