Two optimal double inequalities between power mean and logarithmic mean

For p∈Rp∈R the power mean Mp(a,b)Mp(a,b) of order pp, the logarithmic mean L(a,b)L(a,b) and the arithmetic mean A(a,b)A(a,b) of two positive real values aa and bb are defined by Mp(a,b)={(ap+bp2)1p,p≠0,ab,p=0,L(a,b)={b−alogb−loga,a≠b,a,a=b and A(a,b)=a+b2, respectively.In this article, we answer the questions: What are the greatest values pp and rr, and the least values qq and ss, such that the inequalities Mp(a,b)≤13A(a,b)+23L(a,b)≤Mq(a,b) and Mr(a,b)≤23A(a,b)+13L(a,b)≤Ms(a,b) hold for all a,b>0?a,b>0?