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1
Content available remote

A generalization of the Hahn-Banach theorem

100%
Plewnia J.
Annales Polonici Mathematici
|
1993
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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.
2
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On regularization in superreflexive Banach spaces by infimal convolution formulas

100%
Cepedello-Boiso M.
Studia Mathematica
|
1998
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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
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Uniformly convex functions II

100%
Ma W., Minda D.
Annales Polonici Mathematici
|
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.
4
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Uniformly convex functions

80%
Ma W., Minda D.
Annales Polonici Mathematici
|
1992
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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.
5
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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.
6
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On uniformly convex functions

80%
Goodman A.
Annales Polonici Mathematici
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1991-1992
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tom 56
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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.
7
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Additive inequalities for weighted harmonic and arithmetic operator means

71%
Dragomir S.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2019
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tom 73
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nr 1
EN
In this paper we establish some new upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means under various assumptions for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.
8
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Monotone Valuations on the Space of Convex Functions

71%
Cavallina L., Colesanti A.
Analysis and Geometry in Metric Spaces
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2015
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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.
9
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Fekete-Szegö inequalities associated withkthroot transformation based on quasi-subordination

71%
Magesh N., Yamini J.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2017
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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.
10
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The cancellation law for inf-convolution of convex functions

71%
Zagrodny D.
Studia Mathematica
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1994
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tom 110
|
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.
11
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Coefficient bounds for some subclasses of p-valently starlike functions

61%
Selvaraj C., Babu O. S., Murugusundaramoorthy G.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2013
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tom 67
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nr 2
EN
For functions of the form \[f(z) = z^{p} + \sum_{n = 1}^{\infty} a_{p + n} z^{p + n}\] we obtain sharp bounds for some coefficients functionals in certain subclasses of starlike functions. Certain applications of our main results are also given. In particular, Fekete-Szego-like inequality for classes of functions defined through extended fractional differintegrals are obtained.
12
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Reverse Jensen’s type Trace Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

61%
Dragomir S. S.
Annales Mathematicae Silesianae
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2016
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tom 30
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nr 1
39-62
EN
Some reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest and reverses of Hölder and Schwarz trace inequalities are also given.
13
Content available remote

Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm

41%
Matkowski J., Pycia M.
Annales Polonici Mathematici
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1994-1995
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tom 60
|
nr 3
221-230
EN
Let a and b be fixed real numbers such that 0 < min{a,b} < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsup_{t → 0+} f(t) ≤ 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.
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