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For p&#x02208;&#x211D;, the generalized logarithmic mean Lp(a,b), arithmetic mean A(a,b), and geometric mean G(a,b) of two positive numbers a and b are defined by Lp(a,b)=a, for a=b, Lp(a,b)=[(bp+1-ap+1)/((p+1)(b-a))]1/p, for p&#x02260;0, p&#x02260;-1, and a&#x02260;b, Lp(a,b)=(1/e)(bb/aa)1/(b-a), for p=0, and a&#x02260;b, Lp(a,b)=(b-a)/(log&#x2061;&#x02061;b-log&#x2061;&#x02061;a), for p=-1, and a&#x02260;b, A(a,b)=(a+b)/2, and G(a,b)=ab, respectively. In this paper, we find the greatest value p (or least value q, resp.) such that the inequality Lp(a,b)&#x0003c;&#x003b1;A(a,b)+(1-&#x003b1;)G(a,b) (or &#x003b1;A(a,b)+(1-&#x003b1;)G(a,b)&#x0003c;Lq(a,b), resp.) holds for &#x03B1;&#x2208;(0,1/2)(or &#x03B1;&#x2208;(1/2,1), resp.) and all a,b&#x0003e;0 with a&#x02260;b

Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means