Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

In the article, we present the best possible parameters $\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}\in(0, 1)$ and $\alpha_{3}, \alpha_{4}, \beta_{3}, \beta_{4}\in(0, 1/2)$ such that the double inequalities $$\begin{aligned}& \alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)< X(a,b)< \beta _{1}A(a,b)+(1-\beta_{1})H(a,b), \\ & \alpha_{2}A(a,b)+(1-\alpha_{2})G(a,b)< X(a,b)< \beta _{2}A(a,b)+(1-\beta_{2})G(a,b), \\ & H\bigl[\alpha_{3}a+(1-\alpha_{3})b, \alpha_{3}b+(1- \alpha _{3})a\bigr]< X(a,b)< H\bigl[\beta_{3}a+(1- \beta_{3})b, \beta_{3}b+(1-\beta_{3})a\bigr], \\ & G\bigl[\alpha_{4}a+(1-\alpha_{4})b, \alpha_{4}b+(1- \alpha _{4})a\bigr]< X(a,b)< G\bigl[\beta_{4}a+(1- \beta_{4})b, \beta_{4}b+(1-\beta_{4})a\bigr] \end{aligned}$$ hold for all $a, b>0$ with $a\neq b$ . Here, $X(a,b)$ , $A(a,b)$ , $G(a,b)$ and $H(a,b)$ are the Sandor, arithmetic, geometric and harmonic means of a and b, respectively.