English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For x,y&#x003E;0, &#x02009;a,b&#x2208;&#x211D;, with a+b&#x2260;0, the generalized Muirhead mean M(a,b;x,y) with parameters a and b and the identric mean I(x,y) are defined by M(a,b;x,y)=((xayb+xbya)/2)1/(a+b) and I(x,y)=(1/e)(yy/xx)1/(y&#x2212;x), x&#x2260;y, I(x,y)=x, x=y, respectively. In this paper, the following results are established: (1) M(a,b;x,y)&#x003E;I(x,y) for all x,y&#x003E;0 with x&#x2260;y and (a,b)&#x2208;{(a,b)&#x2208;&#x211D;2:a+b&#x003E;0,&#x02009;ab&#x2264;0,&#x02009;2(a&#x2212;b)2&#x2212;3(a+b)+1&#x2265;0,&#x02009;3(a&#x2212;b)2&#x2212;2(a+b)&#x2265;0}; (2) M(a,b;x,y)&#x003C;I(x,y) for all x,y&#x003E;0 with x&#x2260;y and (a,b)&#x2208;{(a,b)&#x2208;&#x211D;2:a&#x2265;0,&#x02009;b&#x2265;0,&#x02009;3(a&#x2212;b)2&#x2212;2(a+b)&#x2264;0}&#x222A;{(a,b)&#x2208;&#x211D;2:a+b&#x003C;0}; (3) if (a,b)&#x2208;{(a,b)&#x2208;&#x211D;2:a&#x003E;0,&#x02009;b&#x003E;0,&#x02009;3(a&#x2212;b)2&#x2212;2(a+b)&#x003E;0}&#x222A;{(a,b)&#x2208;&#x211D;2:ab&#x003C;0,&#x02009;3(a&#x2212;b)2&#x2212;2(a+b)&#x003C;0}, then there exist x1,y1,x2,y2&#x003E;0 such that M(a,b;x1,y1)&#x003E;I(x1,y1) and M(a,b;x2,y2)&#x003C;I(x2,y2)

Some Comparison Inequalities for Generalized Muirhead and Identric Means