By Whiterhead J. H. C.

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10 23 A Generalisation for Power Series The following result holds [11, Remark 2]. k Theorem 55 Let F : (−r, r) → R, F (x) = ∞ k=0 αk x with αk ≥ 0, k ∈ N. ¯ = (b1 , . . , bn ) are sequences of real numbers such that If ¯ a = (a1 , . . , an ) , b ai bi , a2i , b2i ∈ (−r, r) for any i ∈ {1, . . 84) then one has the inequality: n n F b2i F i=1 2 n a2i ≥ F (ai bi ) i=1 . 85) i=1 Proof. Firstly, let us observe that if x, y ∈ R such that xy, x2 , y 2 ∈ (−r, r) , then one has the inequality [F (xy)]2 ≤ F x2 F y 2 .

N} , then one has the 2. 94) 2 , m > 0. 11. 11 25 A Generalisation of Callebaut’s Inequality The following result holds (see also [11, Theorem 2] for a generalisation for positive linear functionals). k Theorem 56 Let F : (−r, r) → R, F (x) = ∞ k=0 αk x with αk ≥ 0, k ∈ ¯ = (b1 , . . , bn ) are sequences of nonnegative real N. If ¯ a = (a1 , . . , an ) , b numbers such that , a2−α bαi ∈ (0, r) for any i ∈ {1, . . 96) then one has the inequality 2 n n n F aαi b2−α i ≤ F (ai bi ) i=1 F ai2−α bαi .

712:110A. 33 34 BIBLIOGRAPHY [10] T. POPOVICIU, Gazeta Matematicˇ a, 16 (1940), p. 334. S. DRAGOMIR, Inequalities of Cauchy-Buniakowski-Schwartz’s type for positive linear functionals (Romanian), Gaz. Mat. Metod. (Bucharest), 9 (1988), 162-164. [12] T. ANDRESCU, D. O. DRˆIMBE, The trinomial principle in obtaining inequalities (Romanian), Gaz. Mat. (Bucharest), 90 (1985), 332-338. ¨ [13] P. S. , 20 (1965), 136. S. WAGNER, Amer. Math. , Notices, 12 (1965), 220. S. DRAGOMIR, A version of Wagner’s inequality for complex numbers, submitted.