Bernstein-Doetsch type results for (k;h)-convex functions

In this paper we define the so-called (k; h)-convex function which is a natural generalization of the usual convexity, the s-convexity in the first and second sense, the h-convexity, the Godunova-Levin functions and the P-functions. Some regularity and Bernstein-Doetsch type results are investigated for (k; h)-convex functions

On (α,β,a,b)-convex functions

In this paper we investigate the (α,β,a,b)-convex functions which is a common generalization of the usual convexity, the s-convexity in first and second sense, the h-convexity, the Godunova-Levin functions and the P-functions. This notion of convexity was introduced by Maksa and Páles. The main goal of the paper is to prove some regularity and Bernstein-Doetsch type results for (α,β,a,b)-convex functions

Solving functional equations with computer

In this paper we deal with the linear two variable functional equation ℎ_0(x,y)f_0(g_0(x,y))+⋅⋅⋅+ℎ_n(x,y)f_n(g_n(x,y))=F(x,y) where n is a positive integer, g_0, g_1,..., g_n, h_0,ℎ_1,...,ℎ_n and F are given real valued analytic functions on an open set Ω ⊂ ℝ^2,furthermore f_0,f_1,...,f_n are unknown functions. Applying the results of Páles we get recursively an inhomogeneous linear differential-functional equation in one of unknown function for f_1,f_2,...,f_n, respectively. One of our main result states that the solutions of the differential-functional equation obtained are the same as that of an ordinary differential equation (under some assumptions), whose order is usually much smaller than the order of the differetial-functional equation. Our aim is also to describe a computer-program which solves functional equations of this type. This algorithm is implemented in Maple symbolic language

On Hermite-Hadamard type inequalities

The main results of this paper over sucient conditions in order that an approximate lower Hermite-Hadamard type inequality imply an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem

On approximately (k,h)-convex functions

A real valued function $f:D\to \mathbb{R}$ defined on an open convex subset $D$ of a normed space $X$ is called \emph{rationally $(k,h,d)$-convex} if it satisfies $$f\left(k(t)x + k(1-t)y \right) \leq h(t) f(x) + h(1-t) f(y) + d(x,y)$$ for all $x,y\in D$ and $t\in \mathbb{Q} \cap [0,1]$, where $d:X \times X \to \mathbb{R}$ and $k, h:[0,1] \to \mathbb{R}$ are given functions. Our main result is of a Bernstein-Doetsch type. Namely, we prove that (under some natural assumptions) if $f$ is locally bounded from above at a point of $D$ and rationally $(k,h,d)$-convex then it is continuous and $(k,h,d)$-convex

On strongly convex functions

The main results of this paper give a connection between strong Jensen convexity and strong convexity type inequalities. We are also looking for the optimal Takagi type function of strong convexity. Finally a connection will be proved between the Jensen error term and an useful error function

On Strongly Convex Functions

The main results of this paper give a connection between strong Jensen convexity and strong convexity type inequalities. We are also looking for the optimal Takagi type function of strong convexity. Finally a connection will be proved between the Jensen error term and an useful error function

On approximate Hermite-Hadamard type inequalities

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem

Approximate Hermite-Hadamard inequality

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite-Hadamard type inequality imply an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem

Nemsima analízis és alkalmazásai = Nonsmooth analysis and its applications

A Clarke-féle általánosított derivált fogalmat (amely a lokálisan Lipschitz véges dimenziós normált terek között ható függvényekhez társít egy mátrix-halmazértékű deriváltat), általánosítottuk végtelen dimenziós normált tereken értelmezett és duális térként előálló Banach-terekbe képező lokálisan Lipschitz függvényekre, továbbá kidolgoztuk erre az új fogalomra vonatkozó kalkulus szabályok egy teljes spektrumát: összeg-szabály és lánc-szabály, részenként sima függvények differenciálása. Bebizonyítottuk, hogy az így nyert operátor-halmazértékű deriválta legszűkebb szekvenciálisan felülről folytonos halmazértékű szigorú Hadamard-féle prederivált. Vizsgáltuk a konvexitás különböző általánosításait, ezen belül a Jensen-konvex és a t-konvexitás perturbációs tulajdonságait és stabilitását. Ilyen tulajdonságú függvényekre Bernstein-Doetsch-típusú regularitási tételeket igazoltunk. Bevezettük a Q-szubdifferenciál fogalmát, és segítségével a Jensen-konvexitást a Q-szubdifferenciál monotonitásával jellemeztük. A magasabb-rendben Wright-konvex függvényekről megmutattuk, hogy előállnak egy ugyanolyan rendben konvex és egy polinomiális függvény összegeként. Különböző kétváltozós középérték-osztályokban vizsgáltuk az egyenlőségi, homogenitási és összehasonlítási problémákat, valamint az invariancia-egyenlet teljesülésének feltételeit. | Clarke's generalized Jacobian (which assigns a matrix-set-valued derivative to locally Lipschitzian functions acting between finite dimensional normed spaces) was extended to the setting when the domain space is an arbitrary normed space and the range space is a conjugate space of a normed space. The collection of calculus rules (sum and chain rule, derivative for piecewise smooth functions) was also elaborated. It was also shown that this new derivative is the smallest sequentially upper semicontinuous set-valued function which is a Hadamard-type prederivative. Various generalizations of convexity, perturbation and stability properties of convex and monotone functions were investigated. The perturbations of convex, Jensen-convex, and t-convex functions in terms of sums of bounded and Lipschitz functions were described. Regularity theorems of Bernstein-Doetsch type were also obtained. The theory of Q-subdifferential was developed and Jensen-convexity was characterized as a monotonicity property of this subdifferential. The higher-order Wright-convex functions were characterized as the sum of a higher-order convex and a polynomial function. The equality, homogeneity and comparison problem as well as the invariance equation was considered and solved in various classes of two-variable means