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In the article, we provide several sharp bounds for the Toader mean by use of certain combinations of the arithmetic, quadratic, contraharmonic, and Gaussian arithmetic-geometric means.

We present the best possible parameters α1,α2,β1,β2∈R and α3,β3∈(1/2,1) such that the double inequalities α1A(a,b)+(1-α1)C(a,b)0 with a≠b and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, N(a,b), A(a,b), Q(a,b), and C(a,b) are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of a and b, and NQA(a,b)=N[Q(a,b),A(a,b)].

We present the best possible parameters α1,β1,α2,β2∈R and α3,β3∈(1/2,1) such that the double inequalities Qα1(a,b)A1-α1(a,b)0 with a≠b, where A(a,b), Q(a,b), and AG(a,b) are the arithmetic, quadratic, and Gauss arithmetic-geometric means of a and b, respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

We present the best possible parameters λ1,μ1∈R and λ2,μ2∈1/2,1 such that double inequalities λ1C(a,b)+1-λ1A(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/2a2cos2θ+b2sin2θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively.