Abstract
We present the best possible parametersλ1,μ1∈Randλ2,μ2∈1/2,1such that double inequalitiesλ1C(a,b)+1-λ1A(a,b)<T(a,b)<μ1C(a,b)+1-μ1A(a,b),Cλ2a+1-λ2b,λ2b+1-λ2a<T(a,b)<Cμ2a+1-μ2b,μ2b+1-μ2ahold for alla,b>0witha≠b, whereA(a,b)=(a+b)/2,C(a,b)=a3+b3/a2+b2andT(a,b)=2∫0π/2a2cos2θ+b2sin2θdθ/πare the arithmetic, second contraharmonic, and Toader means ofaandb, respectively.
Highlights
B > 0 the Toader mean T(a, b) [1], second contraharmonic mean C(a, b), and arithmetic mean A(a, b) of a and b are given by T (a, b) = 2 π π/2 ∫ √a2cos2θ + b2sin2θ dθ {{{{{{{{{2aE (√1 − π (b/a)2), a > b
We present the best possible parameters λ1, μ1 ∈ R and λ2, μ2 ∈ (1/2, 1) such that double inequalities λ1C(a, b) + (1 − λ1)A(a, b) < T(a, b) < μ1C(a, b) + (1 − μ1)A(a, b), C[λ2a + (1 − λ2)b, λ2b + (1 − λ2)a] < T(a, b) < C[μ2a + (1 − μ2)b, μ2b + (1 − μ2)a] hold for all a, b > 0 with a ≠ b, where A(a, b) = (a + b)/2, C(a, b) = (a3 + b3)/(a2 + b2) and T(a, b) = 2 ∫0π/2 √a2cos2θ + b2sin2θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively
It is well known that Mp(a, b), Gp(a, b), Lp(a, b), and Sq(a, b) are continuous and strictly increasing with respect to p ∈ R and q ∈ [0, 1] for fixed a, b > 0 with a ≠ b, respectively
Summary
We present the best possible parameters λ1, μ1 ∈ R and λ2, μ2 ∈ (1/2, 1) such that double inequalities λ1C(a, b) + (1 − λ1)A(a, b) < T(a, b) < μ1C(a, b) + (1 − μ1)A(a, b), C[λ2a + (1 − λ2)b, λ2b + (1 − λ2)a] < T(a, b) < C[μ2a + (1 − μ2)b, μ2b + (1 − μ2)a] hold for all a, b > 0 with a ≠ b, where A(a, b) = (a + b)/2, C(a, b) = (a3 + b3)/(a2 + b2) and T(a, b) = 2 ∫0π/2 √a2cos2θ + b2sin2θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively. + 2 b, respectively, where E(r) = ∫0π/2(1 − r2sin2t)1/2dt (r ∈ (0, 1)) is the complete elliptic integral of the second kind. It is well known that Mp(a, b), Gp(a, b), Lp(a, b), and Sq(a, b) are continuous and strictly increasing with respect to p ∈ R and q ∈ [0, 1] for fixed a, b > 0 with a ≠ b, respectively.
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