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This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f ∈ Δ (2) , the set of convex functions, from W r , the space of functions on [ - 1 , 1 ] for which f (r - 1) is absolutely continuous and ‖ f (r) ‖ ∞ : = ess sup x ∈ [ - 1 , 1 ] | f (r) (x) | < ∞ : For any f ∈ W r ∩ Δ (2) , r ∈ N , there exists a number N = N (f , r) , such that for every n ≥ N , there is an algebraic polynomial of degree ≤ n which is in Δ (2) and such that f - P n φ r ∞ ≤ c (r) n r ‖ f (r) ‖ ∞ , where φ (x) : = 1 - x 2 . For r = 1 and r = 2 , the above result holds with N = 1 and is well known. For r ≥ 3 , it is not true, in general, with N independent of f. [ABSTRACT FROM AUTHOR]

Let X be a real linear space and $D\subseteq X$ be a nonempty convex subset. Given an error function E:[0,1]×( D− D)→ℝ∪{+∞} and an element $t\in\left]0,1\right[$, a function f: D→ℝ is called ( E, t) -convex if for all x, y∈ D. The main result of this paper states that, for all a, b∈(ℕ∪{0})+{0, t,1− t} such that { a, b, a+ b}∩ℕ≠∅, every ( E, t)-convex function is also $\big(F,\frac{a}{a+b}\big)$-convex, where As a consequence, under further assumptions on E, the strong and approximate convexity properties of ( E, t)-convex functions can be strengthened. [ABSTRACT FROM AUTHOR]

Let $${\mathbb{F}}$$ be a subfield of $${\mathbb{R}}$$ and X be a linear space over $${\mathbb{F}}$$ . Let $${ D\subseteq X }$$ be a nonempty $${\mathbb{F}}$$ -convex set, $${ D^*:=D-D:=\{x-y : x,y\in D\} }$$ , and $${\alpha \colon {D^* \rightarrow \mathbb{R}}}$$ be a nonnegative even function. The function $${ f \colon {D\rightarrow \mathbb{R}}}$$ is called $${(\alpha,\mathbb{F})}$$ -convex, if it satisfies the inequality for all $${x,y\in D}$$ and for all $${t\in \mathbb{F}\cap [0,1]}$$ . In this paper we characterize $${(\alpha,\mathbb{F})}$$ -convex functions by comparison of modified difference ratios and support properties. If $${\alpha}$$ satisfies some additional conditions, we obtain the differentiability of $${(\alpha,\mathbb{F})}$$ -convex functions in the appropriate sense. [ABSTRACT FROM AUTHOR]

Let $${\mathcal S}$$ denote the class of all functions of the form $${f(z)=z+a_2z^2+a_3z^3+\cdots}$$ which are analytic and univalent in the open unit disk $${{\mathbb{D}} }$$ and, for $${\lambda > 0}$$ , let $${\Phi_\lambda (n,f)=\lambda a_n^2-a_{2n-1}}$$ denote the generalized Zalcman coefficient functional. Zalcman conjectured that if $${f\in \mathcal S}$$ , then $${|\Phi_1 (n,f)|\leq (n-1)^2}$$ for $${{n\ge 3}}$$ . The functional of the form $${\Phi_\lambda (n,f)}$$ is indeed related to Fekete-Szegő functional of the $${n}$$ -th root transform of the corresponding function in $${\mathcal S}$$ . This conjecture has been verified for a certain special geometric subclasses of $${\mathcal S}$$ but it remains open for $${f\in {\mathcal S}}$$ and for $${n > 6}$$ . In the present paper, we prove sharp bounds on $${|\Phi_\lambda (n,f)|}$$ for $${f\in \mathcal{F}(\alpha )}$$ and for all $${n\geq 3}$$ , in the case that $${\lambda}$$ is a positive real parameter, where $${ \mathcal{F}(\alpha )}$$ denotes the family of all functions $${f\in {\mathcal S}}$$ satisfying the condition where $${-1/2\leq \alpha < 1}$$ . Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $${\alpha}$$ , $${\alpha \in [-1/2,1)}$$ . [ABSTRACT FROM AUTHOR]

We study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of such integrals. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to prove that Riemannian manifolds that admit monotone vector fields that satisfy a weaker condition than strictly monotone has infinite volume, in a sense, generalizing a result that Bishop and O'Neill proved for convex functions. [ABSTRACT FROM AUTHOR]

Hubert, Mauduit and Sárközy introduced the pseudorandom measure of order ℓ of binary lattices. This measure studies the pseudorandomness only on box lattices of very special type. In certain applications one may need measures covering a more general situation. In this paper the line measure and the convex measure are introduced. [ABSTRACT FROM AUTHOR]

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E n( f) and E ( f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En ( f) ≦ E ( f), and Lorentz and Zeller proved that the inverse inequality E ( f) ≦ cE n( f) is invalid even with the constant c = c( f) which depends on the function f ∈ Δ2. In this paper we prove, for every α > 0 and function f ∈ Δ2, that where c( α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases s≦ 1 and s ≧ 2. [ABSTRACT FROM AUTHOR]

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented. [ABSTRACT FROM AUTHOR]

We present various properties of the Erlang loss function B (x , a) = (a ∫ 0 ∞ e - a t (1 + t) x d t) - 1 (x ≥ 0 , a > 0). Among other results, we prove: The function x ↦ B (x , a) λ is convex on [ 0 , ∞) for every a > 0 if and only if λ ≤ 0 or λ ≥ 1 . The function x ↦ (1 - B (1 / x , a)) - 1 is strictly convex on (0 , ∞) . This leads to the functional inequality 2 1 - B (H (x , y) , a) < 1 1 - B (x , a) + 1 1 - B (y , a) (x , y > 0 , x ≠ y) , where H (x , y) = 2 x y / (x + y) denotes the harmonic mean of x and y. Let a > 0 . The inequality B (x , a) + B (1 / x , a) ≤ 1 holds for all x > 0 if and only if a ≤ 1 . [ABSTRACT FROM AUTHOR]

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings. [ABSTRACT FROM AUTHOR]

We construct a simple volume-preserving map from the cube [0,b]3 to the tetrahedron {(x,y,z)∈R3, x≥0, y≥0, z≥0, x+y+z≤a}, with a=b63. This map will allow us to construct equal-volume subdivisions of arbitrary tetrahedrons and arbitrary convex polyhedrons into polyhedral cells. Moreover, mapping the regular octahedron onto the ball using a volume-preserving map previously constructed by the authors, one can obtain uniform and refinable grids on the 3D ball by a simple procedure, starting from appropriate grids on the cube. [ABSTRACT FROM AUTHOR]

For strictly increasing concave functions φ whose inverse functions are log-concave, the φ-Brunn-Minkowski inequality for planar convex bodies is established. It is shown that for convex bodies in Rn the φ-Brunn-Minkowski is equivalent to the φ-Minkowski mixed volume inequalities. [ABSTRACT FROM AUTHOR]

For polyhedral convex cones in Rd, we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type. [ABSTRACT FROM AUTHOR]

Let X be a real Hilbert space. We give necessary and sufficient algebraic conditions for a mapping $${F\colon X \to X}$$ with a closed image set to be the metric projection mapping onto a closed convex set. We provide examples that illustrate the necessity of each of the conditions. Our characterizations generalize several results related to projections onto closed convex sets. [ABSTRACT FROM AUTHOR]

We give explicit formulas and algorithms for the computation of the Thurston-Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds. [ABSTRACT FROM AUTHOR]

We show by a direct construction that there are at least exp $${\{cV^{(d-1)/(d+1)}\}}$$ convex lattice polytopes in $${\mathbb{R}^d}$$ of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family $${\mathcal{P}^{d}(r)}$$ (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r/ d!. Namely we prove that for $${P \in \mathcal{P}^{d}(r), d!}$$ vol P takes all possible integer values between cr and r where $${c > 0}$$ is a constant depending only on d. [ABSTRACT FROM AUTHOR]

We show by a construction that there are at least exp { cV} convex lattice polytopes in ℝ of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. [ABSTRACT FROM AUTHOR]

Properties of Fourier-Haar coefficients of continuous functions are studied. It is established that Fourier-Haar coefficients of continuous functions are monotonic in a certain sense for convex functions. Questions of quasivariation of Fourier-Haar coefficients of continuous functions are also considered. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet operator H=− d/ dx+ q( x) on L([ t,∞)), where q is a convex potential with q( x)→∞ as x→∞. We show that the eigenvalue gap Γ( t) of H is monotone increasing as t increases from −∞ to ∞. We also show that Γ( t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of Γ( t) for quadratic potentials is obtained. [ABSTRACT FROM AUTHOR]

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented. [ABSTRACT FROM AUTHOR]

It is proved that if f is continuous and the approximate symmetric d.l.V.P. derivatives D a n-2 f of f of order n-2 exist in ( a,b) then under a certain smoothness type condition on f, D a n-2 f is in Baire*1. Also Zahorski property and Denjoy property for the ordinary symmetric d.l.V.P. derivative D n f are established under certain suitable conditions. [ABSTRACT FROM AUTHOR]

Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations. [ABSTRACT FROM AUTHOR]

We present some relations for inequalities involving certain differential-integral operators. [ABSTRACT FROM AUTHOR]