Geodesic
Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications
Communications in Mathematics, Tome 22 (2014) no. 2, pp. 107-132

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Some new bounds for the Čebyšev functional in terms of the Lebesgue norms $$ \biggl \Vert f-\frac {1}{b-a}\int _a^b f(t){\,\mathrm{d}t} \biggr \Vert _{[a,b],p} $$ and the $\Delta $-seminorms $$ \lVert f\rVert _{p}^{\Delta } := \biggl (\int _a^b \int _a^b |f(t)-f(s)|^{p}{\,\mathrm{d}t} {\,\mathrm{d}s} \biggr )^{\frac 1p} $$ are established. Applications for mid-point and trapezoid inequalities are provided as well.
Classification : 25D10, 26D15
Keywords: Absolutely continuous functions; Convex functions; Integral inequalities; Čebyšev functional; Jensen's inequality; Lebesgue norms; Mid-point inequalities; Trapezoid inequalities 
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     author = {Dragomir, Silvestru Sever},
     title = {Bounds for {Convex} {Functions} of {\v{C}eby\v{s}ev} {Functional} {Via} {Sonin's} {Identity} with {Applications}},
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     pages = {107--132},
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Dragomir, Silvestru Sever. Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications. Communications in Mathematics, Tome 22 (2014) no. 2, pp. 107-132. http://geodesic.mathdoc.fr/item/COMIM_2014__22_2_a0/