# (M,N) - konveksne funkcije pridružene paru sredina M i N

Ilić, Matija (2015) (M,N) - konveksne funkcije pridružene paru sredina M i N. Diploma thesis, Faculty of Science > Department of Mathematics.

 Preview
PDF
Language: Croatian

Archive (dodatni materijali)
Language: Croatian

Viewed through history of mathematics, area of $(M,N)$ - convex functions is relatively young. The first works on the theme of convex functions appeared in the late 19th century, and for the pioneer of this field of mathematics is considered Danish mathematician J.L.W.V. Jensen. $(M,N)$ - convex function have begun to be studied only in the 20th century, and today it is generally accepted the following definition. Let $f:I\rightarrow \R^+$ be continuous, where $I$ is a subinterval of $\R^+$, and let $M$ and $N$ be any two mean functions. We say that $f$ is $\left(M,N\right)$-convex if for all $x,y\in I$: $$f\left(M\left(x,y\right)\right)\leq N\left(f\left(x\right),f\left(y\right)\right).$$ In the first section of this paper we have process the concept of convex function and gave several ways of defining them. We have also expressed and proved some important theorems for our main part of the work, such as the Hermite - Hadamard theorem. We concluded chapter by stating the properties of convex functions which we will use later. In Chapter 2 we reminded ourselves of the known inequalities between the harmonic, geometric, arithmetic and square means. Below, we defined the weighted mean and the weighted mean of order t. At the end of the second chapter we expressed and proved the so-called Theorem of monotony. The main parts of this work are Chapters 3 and 4. In Chapter 3 we defined $(M,N)$ - convex functions and studied the criteria and properties of convexity of functions according to the pair of means, where for means we have taken harmonic, geometric and arithmetic mean. After that we proved several theorems and propositions for $(M,N)$ - convex functions related to the quasi-arithmetic means and the means of order t. The last chapter of this work is devoted to $(A,G)$ and $(G,A)$ - convex functions. We defined the gamma and beta functions, logarithmic mean and proved various theorems related to $(A,G)$ and $(G,A)$ - convex functions.