  For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the
inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$
and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for
all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq 1/2-\sqrt{2s}/(4s)$ and
$t_{2}\geq 1/2-\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$
are the geometric, logarithmic, arithmetic-geometric and arithmetic means of
$a$ and $b$, respectively.
