In the present paper, we show that if A ∈Mn(C) is a non scalar strictly positive matrix such that 1 ∈ σ(A) , and p > q > 1 with p + q = 1, then there exists X ∈ Mn(C) such that ω(AXA) > ω( p A pX + q XA q) . Moreover, several numerical radius inequalities are presented for Hilbert space operators. In particular, we prove that if p q > 1 with p + 1 q = 1 , then ω r(A∗XB) ∥ ∥ ∥ 1 p (A ∗|X∗|A) rp 2 + q (B∗|X |B) rq 2 ∥ ∥ ∥ , for all A,B,X ∈ B(H) and r q . Mathematics subject classification (2010): 15A60, 15A42, 47A30.