Matematiqki Bilten ISSN 0351-336X Vol.38 (LXIV) No.2 2014 (53–68) UDC: 517.518.28:517.417.124 Skopje, Makedonija GENERALISATIONS OF STEFFENSEN’S INEQUALITY BY HERMITE’S POLYNOMIAL JULIJE JAKŠETIĆ, JOSIP PEČARIĆ, AND ANAMARIJA PERUŠIĆ Abstract. We study generalizations of Steffensen’s inequality using Hermite expansions with integral reminder. In comparing differences of two weighted integrals we vary on the number of knots in expansion which leads us to gen- eralization of conditions for Steffensen’s inequality. After that, we construct exponentially convex functions and Cauchy means. 1. Introduction Let −∞ <α<b< ∞, and a ≤ a 1 <a 2 ... < a r ≤ b, (r ≥ 2) be given. For f ∈ C n [a,b] a unique polynomial P H (t) of degree (n − 1), exists, fulfilling one of the following conditions: Hermite conditions: P (i) H (a j )= f (i) (a j ); 0 ≤ i ≤ k j , 1 ≤ j ≤ r, r  j=1 k j + r = n, in particular: Simple Hermite or osculatory conditions: (n =2m, r = m, k j =1 for all j ) P O (a j )= f (a j ),P ′ O (a j )= f ′ (a j ), 1 ≤ j ≤ m, Lagrange conditions: (r = n, k j =0 for all j ) P L (a j )= f (a j ), 1 ≤ j ≤ n, Type (m,n − m) conditions: (r =2, 1 ≤ m ≤ n−1, k 1 = m−1,k 2 = n−m−1) P (i) mn (a)= f (i) (a), 0 ≤ i ≤ m − 1, P (i) mn (b)= f (i) (b), 0 ≤ i ≤ n − m − 1, One-point Taylor conditions: (r =1,k 1 = n − 1) P (i) T (a)= f (i) (a), 0 ≤ i ≤ n − 1. 2010 Mathematics Subject Classification. Primary: 26D15, Secondary: 26A51. Key words and phrases. Steffensen’s inequality, Hermite polynomial, n-exponential convex- ity, n-convexity . 53