Let A,B,X ∈Mn(C) and ||| · ||| be an arbitrary unitarily invariant norm. We give a new function f (t, s) that is log-convex in each of its variables such that f (1/2, 1/2) ≤ f (t, s) for any t, s ∈ [0, 1] which generalize the log-convex function defined in [4] and obtain the inequalities as follows: |||AXB∗|||2 = f (1/2, 1/2) ≤ f (t, 1 − t) ≤ (t|||A∗AX||| + (1 − t)|||XB∗B||| − r(√|||A∗AX||| − √|||XB∗B|||)2) × ((1 − t)|||A∗AX||| + t|||XB∗B||| − r(√|||A∗AX||| − √|||XB∗B|||)2), where t ∈ [0, 1] and r = min {t, 1 − t} . Furthermore, we refine some inequalities as well