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• W3201788911 abstract "Abstract In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A 1 , …, A n be bounded linear operators on a complex separable Hilbert space <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow class=MJX-TeXAtom-ORD> <m:mi class=MJX-tex-caligraphic mathvariant=script>H</m:mi> </m:mrow> </m:math> $mathcal{H}$ and let α 1 , …, α n be positive real numbers such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:munderover> <m:mo movablelimits=false>∑</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> $sumlimits^{n}_{j=1}alpha_{j}=1$ . We show that for every unitarily invariant norm, If f is a non-negative function on [0, ∞) such that f (0) = 0 and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>g</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:msqrt> <m:mi>t</m:mi> </m:msqrt> <m:mo stretchy=false>)</m:mo> </m:math> $g(t)=f(sqrt{t})$ is convex, then <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mtable columnalign=right left right left right left right left right left right left rowspacing=3pt columnspacing=0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em> <m:mtr> <m:mtd> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:munderover> <m:mo movablelimits=false>∑</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mo stretchy=false>)</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> </m:mtd> <m:mtd> <m:mo>≥</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:munder> <m:mo movablelimits=false>∑</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>S</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> </m:mrow> </m:munder> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>(</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>(</m:mo> </m:mrow> <m:msqrt> <m:mfrac> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mn>4</m:mn> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> <m:mo stretchy=false>(</m:mo> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mfrac> </m:msqrt> <m:mspace width=thickmathspace /> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:munderover> <m:mo movablelimits=false>∑</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>)</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd /> <m:mtd> <m:mspace width=2em /> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>(</m:mo> </m:mrow> <m:msqrt> <m:mfrac> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo stretchy=false>(</m:mo> <m:mn>2</m:mn> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mrow> <m:mn>4</m:mn> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> <m:mo stretchy=false>(</m:mo> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>ℓ</m:mi> </m:mrow> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mfrac> </m:msqrt> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow class=MJX-TeXAtom-ORD> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>)</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>(</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:munderover> <m:mo movablelimits=false>∑</m:mo> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mi>α</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>)</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mo maxsize=2.047em minsize=2.047em>|</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:math> $$begin{align*}bigg|bigg|bigg|sumlimits^{n}_{j=1}alpha_{j}f(|A_{j}|)bigg|bigg|bigg|&geqbigg|bigg|bigg|sumlimits_{j,kin S_{ell}}bigg(fbigg(sqrt{frac{alpha_{j}alpha_{k}}{4alpha_{ell}(1-alpha_{ell})}};bigg|A_{j}+A_{k}-2sumlimits^{n}_{j=1}alpha_{j}A_{j}bigg|bigg)&qquad+fbigg(sqrt{frac{alpha_{j}alpha_{k}(2alpha_{ell}-1)}{4alpha_{ell}(1-alpha_{ell})}}|A_{j}-A_{k}|bigg)bigg)+fbigg(bigg|sumlimits^{n}_{j=1}alpha_{j}A_{j}bigg|bigg)bigg|bigg|bigg|end{align*}$$ for ℓ = 1, …, n . If f is a non-negative function on [0, ∞) such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>g</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:msqrt> <m:mi>t</m:mi> </m:msqrt> <m:mo stretchy=false>)</m:mo> </m:math> $g(t)=f(sqrt{t})$ is concave, then the inverse inequality holds. Here, the symbol S ℓ = {1, …, n } ∖ {ℓ} for ℓ ∈ {1, …, n }. In addition, we provide some applications of the above inequalities." @default.
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• W3201788911 title "Clarkson inequalities related to convex and concave functions" @default.
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