Affiliation:
1. ATATÜRK ÜNİVERSİTESİ, FEN BİLİMLERİ ENSTİTÜSÜ
Abstract
We present the generalizations of Hölder's inequality and Minkowski's inequality along with the generalizations of Aczel's, Popoviciu's, Lyapunov's and Bellman's inequalities. Some applications for the metric spaces, normed spaces, Banach spaces, sequence spaces and integral inequalities are further specified. It is shown that $({\mathbb{R}}^n,d)$ and $\left(l_p,d_{m,p}\right)$ are complete metric spaces and $({\mathbb{R}}^n,{\left\|x\right\|}_m)$ and $\left(l_p,{\left\|x\right\|}_{m,p}\right)$ are $\frac{1}{m}-$Banach spaces. Also, it is deduced that $\left(b^{r,s}_{p,1},{\left\|x\right\|}_{r,s,m}\right)$ is a $\frac{1}{m}-$normed space.
Publisher
Mathematical Sciences and Applications E-Notes
Reference40 articles.
1. [1] Beckenbach, E .F., Bellman, R.: Inequalities, Springer-Verlag, Berlin (1961).
2. [2] Royden, H. L.: Real analysis. Macmillan Publishing Co. Inc. New-York (1968).
3. [3] Yosida, K.: Functional analysis. Springer-Verlag Berlin, Heidelberg, New-York (1974).
4. [4] Bi¸sgin, M. C.: The binomial sequence spaces which include the spaces lp and l1 and geometric properties. J. Inequal. Appl.2016, 304 (2016).
5. [5] Ellidokuzo˘ glu, H. B., Demiriz, S., Köseo˘ glu, A.: On the paranormed binomial sequence spaces. Univers. J. Math. Appl. 1, 137-147 (2018).