On approximating the quasi-arithmetic mean

In this article, we prove that the double inequalities $${aligned} &α₁ [7C(a,b)/16+9H(a,b)/16 ]+(1- α₁) [3A(a,b)/4+G(a, b)/4 ]\\ &< E(a,b) \\ &< β₁ [7C(a,b)/16+9H(a,b)/16 ]+(1- β₁) [3A(a,b)/4+G(a, b)/4 ], \\ &[7C(a,b)/16+9H(a,b)/16 ]^{α ₂} [ 3A(a,b)/4+G(a, b)/4 ]^{1-α₂}\\ &< E(a,b) \\ &< [7C(a,b)/16+9H(a,b)/16 ]^{β ₂} [ 3A(a,b)/4+G(a, b)/4 ]^{1-β₂} {aligned}$$ hold for all $a, b>0$ with $a≠ b$ if and only if $α₁≤ 3/16=0.1875$ , $β₁≥64/π²-6= 0.484555$ , $α₂≤3/16=0.1875$ and $β₂≥(5log2-log3-2log π)/(log7-log6)= 0.503817$ , where $E(a,b)= (2/π∫π/2₀√{acos²θ +bsin²θ}\,dθ )²$ , $H(a,b)=2ab/(a+b)$ , $G(a,b)=√ab$ , $A(a,b)=(a+b)/2$ and $C(a,b)=(a²+b²)/(a+b)$ are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively.