Exactness of a nontrivial estimate in a cyclic inequality

1976 ◽  
Vol 20 (2) ◽  
pp. 673-675 ◽  
Author(s):  
E. K. Godunova ◽  
V. I. Levin
Keyword(s):  
Nontrivial Estimate ◽  
Cyclic Inequality

scholarly journals ISOPERIMETRIC INEQUALITIES FOR Lp GEOMINIMAL SURFACE AREA

2011 ◽  
Vol 53 (3) ◽  
pp. 717-726 ◽  
Author(s):  
BAOCHENG ZHU ◽  
NI LI ◽  
JIAZU ZHOU
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Type Inequality ◽  
Isoperimetric Inequalities ◽  
Surface Areas ◽  
Cyclic Inequality ◽  
Geominimal Surface Area

AbstractIn this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

On a cyclic inequality

1973 ◽  
Vol 4 (2-3) ◽  
pp. 163-168 ◽  
Author(s):  
Á. Elbert
Keyword(s):  
Cyclic Inequality

scholarly journals On an Extension of Shapiro's Cyclic Inequality

2009 ◽  
Vol 2009 (1) ◽  
pp. 491576
Author(s):  
NguyenMinh Tuan ◽  
LeQuy Thuong
Keyword(s):  
Cyclic Inequality

scholarly journals Cyclic Inequality Forms with Power 1/2,1/3

2017 ◽  
Vol 33 (3) ◽  
Author(s):  
Quốc Phạm Văn
Keyword(s):  
Real Numbers ◽  
Positive Real ◽  
Cyclic Inequality

The purpose of this paper is to establish inequalities between two terms \begin{equation*} F =\sum_{i=1}^{n}\sqrt{ax_{i}^{2}+bx_{i}x_{i+1}+cx_{i+1}^{2}+dx_i+ex_{i+1}+d }; \end{equation*} \begin{equation*} G =\sum_{i=1}^{n}\sqrt[3]{ax_{i}^{3}+bx_{i}^{2}x_{i+1}+cx_{i}x_{i+1}^{2}+dx_{i+1}^{3}}, \end{equation*} and $\sum_{i=1}^{n}x_{i}$ for a sequence of cyclic positive real numbers $ (x_{i})_{i=1}^{n+1}$ with $x_{n+1}=x_{1}$. The results depends on the sign of expressions containing the coefficients $a,b,c,d$. The general case for $F$ is also investigated.

scholarly journals A Cyclic Inequality and an Extension of it. II

1962 ◽  
Vol 13 (2) ◽  
pp. 143-152 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Real Numbers ◽  
Positive Real ◽  
Cyclic Inequality ◽  
Image Position ◽  
Positive Integers

Throughout this paper, unless otherwise stated, n and L stand for positive integers and α, t, x, x1, x2, … for positive real numbers. Letwhereand

scholarly journals ELEMENTS OF HIGH ORDER ON FINITE FIELDS FROM ELLIPTIC CURVES

2010 ◽  
Vol 81 (3) ◽  
pp. 425-429 ◽  
Author(s):  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Rational Functions ◽  
Large Degree ◽  
High Order ◽  
Prime Field ◽  
Nontrivial Estimate ◽  
Special Cases

AbstractWe discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

scholarly journals Double Character Sums over Subgroups and Intervals

2014 ◽  
Vol 90 (3) ◽  
pp. 376-390 ◽  
Author(s):  
MEI-CHU CHANG ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Upper Bound ◽  
Multiplicative Group ◽  
Character Sums ◽  
Multiplicative Character ◽  
Nontrivial Estimate ◽  
Image Position ◽  
Primitive Roots ◽  
Double Character

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

26.—On Shapiro's Cyclic Inequality

1976 ◽  
Vol 75 (4) ◽  
pp. 333-338 ◽  
Author(s):  
P. J. Bushell ◽  
A. H. Craven
Keyword(s):  
Numerical Analysis ◽  
Elementary Proof ◽  
Negative Numbers ◽  
Counter Example ◽  
Cyclic Inequality

SynopsisThe inequality, for suitable sets of non-negative numbers x1, x2, …, xn,is undecided in the cases n = 11, 12, 13, 15, 17, 19, 21 and 23. In this note we present the results of numerical analysis supporting the conjecture that the inequality is valid in these cases. A new counter–example for n = 25 and a new elementary proof when n = 7 are given.

scholarly journals On Shapiro’s cyclic inequality for $N=13$

1985 ◽  
Vol 45 (171) ◽  
pp. 199-199
Author(s):  
B. A. Troesch
Keyword(s):  
Cyclic Inequality
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