Jensen's inequality

[MusaAli]

ok so here's a sample question:

"The 1‐year spot rate is i1, the 4‐year spot rate is i4, and the forward rate between years 1 and 4 is r0(1,4). Using Jensen’s inequality, formulate an inequality comparing i4 to a*i1 + b*r0(1,4), including the values of a and b and the direction of the inequality."

and here's the author's solution:

"(1+i1)*(1+r0(1,4))3 = (1+i4)4. Therefore, taking logarithms of both sides we determine that ¼*ln(1+i1) + ¾*ln(1+r0(1,4)) = ln(1+i4). However, ln(1+x) is a concave function, so using Jensen’s inequality,
ln(1+i4) = ¼*ln(1+i1) + ¾*ln(1+r0(1,4)) <= ln(1 + ¼*i1 + ¾*r0(1,4)). Since ln(1+x) is also a monotone increasing function, this means that i4 <= ¼*i1 + ¾*r0(1,4)."

Now here's what i don't get, i thought jensen said: f(E[x]) <= E[f(x)] so isn't the author doing exactly OPPOSITE? shouldn't it be:
f(E[x]) = ln(1 + ¼*i1 + ¾*r0(1,4))
and
E[f(x)] = ¼*ln(1+i1) + ¾*ln(1+r0(1,4))
meaning:
ln(1 + ¼*i1 + ¾*r0(1,4)) <= ¼*ln(1+i1) + ¾*ln(1+r0(1,4)) ???

[Abraham Weishaus]

logarithm is a concave, not convex, function, so Jensen's inequality is reversed.

[MusaAli]

thanks a ton =D