We shall state the following parametric extensions of Shannon inequality and its reverse one in Hilbert space operators. Let p∈[0,1] and also let { A 1, A 2,…, A n } and { B 1, B 2,…, B n } be two sequences of strictly positive operators on a Hilbert space H such that ∑ j=1 n A j ♯ p B j ⩽ I . Then ∑ j=1 nS p+1(A j|B j)⩾ ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾ log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p(A j|B j) ⩾− log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾− ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p−1(A j|B j), where S q(A|B)=A 1 2 A −1 2 BA −1 2 q logA −1 2 BA −1 2 A 1 2 for A>0 , B>0 and any real number q and A♮ qB=A 1 2 A −1 2 BA −1 2 qA 1 2 for A>0 , B>0 and any real number q . In particular, if ∑ j=1 n A j =∑ j=1 n B j = I , then ∑ j=1 nS 2(A j|B j)⩾ ∑ j=1 nB jA j −1B j log ∑ j=1 nB jA j −1B j ⩾ log ∑ j=1 nB jA j −1B j ⩾∑ j=1 nS 1(A j|B j)⩾0 ⩾∑ j=1 nS(A j|B j)⩾− log ∑ j=1 nA jB j −1A j ⩾− ∑ j=1 nA jB j −1A j log ∑ j=1 nA jB j −1A j ⩾∑ j=1 nS −1(A j|B j), where S(A|B)=S 0(A|B)=A 1 2 logA −1 2 BA −1 2 A 1 2 which is the relative operator entropy of A>0 and B>0 . Our results can be considered as parametric extensions of the following celebrated Shannon inequality [Ann. Math. Statistics 22 (1951) 79; Bull. Syst. Tech. J. 27 (1948) 379; Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman, 1998, p. 233] which is very useful and so famous in information theory. Let { a 1, a 2,…, a n } and { b 1, b 2,…, b n } be two probability vectors. Then 0⩾∑ j=1 n a j log b j −∑ j=1 n a j log a j (see inequalities (2.4) of Corollary 2.4).