ON LANDAU S THEOREMS. 1. Introduction E. Landau has proved the following theorems [11]:

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1 GLASNIK MATEMATIČKI Vol. 39(59)(004), ON LANDAU S THEOREMS Dragoslav S. Mitrinović, Josip E. Pečarić and Hrvoje Kraljević University of Belgrade, Yugoslavia and University of Zagreb, Croatia Abstract. In this paper we give some applications and special cases of a generalization of the Landau s theorem for Frechet-differentiable functions. 1. Introduction E. Landau has proved the following theorems [11]: Theorem A. Let I R be an interval of length not less than and let f : I R be a twice differentiable function satisfying f(x) 1 and f (x) 1 (x I). Then f (x) (x I). Furthermore, is the best possible constant in the above inequality. Theorem B. Let f : R R be a twice differentiable function satisfying f(x) 1 and f (x) 1 (x I). Then f (x) (x R). Furthermore, is the best possible constant in the above inequality. There exists many generalizations of these results. In Section we give some remarks about the generalization of Theorem A given in [9]. Some applications and special cases are given in Section Mathematics Subject Classification. 6D10, 47D05, 47D10. Key words and phrases. Differentiable functions, Frechet-differentiability, Landau s inequalities. 57

2 58 D.S. MITRINOVIĆ, J.E. PEČARIĆ AND H. KRALJEVIĆ by. Landau s theorems for Frechet-differentiable functions Let X and Y be Banach spaces. Given a, b X (a b) define g : X R g(x) = x a + b x (x X). Let D be a convex subset of X such that g(x) b a for every x D and suppose that a, b D. Furthermore, let f : X Y be twice Frechet differentiable on D. With these assumptions the following generalizations of Theorems A and B have been proven in [9]: Theorem C. If F (x) M (x D) and F (x) (h, h) N h (h X, x D), then F(x) (b a) M + N g(x) M + N b a (x D). Theorem D. If F (x) (h, h) N h (h X, x D), then F (x) (b a) F (b) + F (a) N g(x) (x D). We prove now the following generalization of these results. Theorem.1. Suppose that (.1) F(x) (h, h) H(h) (h X, x D), where H is a function from X to R +. Then for all x D (.) F (x) (b a) F (b) + F (a) 1 (H(a x) + H(b x)). Under the further assumption (.3) F (x) M (x D), then for all x D (.4) F(x) (b a) M + 1 (H(a x) + H(b x)). Proof. If x D and h X are such that x + th D for every t, 0 < t < 1, then the Taylor s formula holds true: F (x + h) = F (x) + F (x) (h) + w(x, h) where w(x, h) = 1 F (x+th)(h, h) for some t, 0 < t < 1. Combining the two formulas for h = a x and h = b x we obtain (.5) F (x) (b a) F (b) + F (a) = w(x, a x) w(b x). Now, (.5) together with (.1) implies (.). Similarly, (.5) together with (.1) and (.3) implies (.4).

3 ON LANDAU S THEOREMS 59 Remark.. If the function H in Theorem.1 is even (H( h) = H(h)) then for a = x h and b = x + h we obtain from (.4): (.6) F(x) (h) M + H(h). Remark.3. The inequalities (.4) and (.6) hold true if instead of (.3) we have (.7) F (b) F (a) M. Remark.4. For (H(h) = N h we obtain Theorems C and D. 3. Some applications Corollary 3.1. Let f : [a, b + h] R be a differentiable function (a < b, h > 0) such that (3.8) δ h f (x) N (x (a, b)), where δ h g(x) = 1 h (g(x + h) g(x)). Then (3.9) (b a)δ h f(x) f(b) + f(a) N [ (x a) + (x b) ]. If we also have (3.10) m f(x) M (a x b + h), then (3.11) (b a) δ h f(x) M m + N [ (x a) + (x b) ]. Proof. This follows from Theorem.1 and Remark.3 for X = Y = R, x = x, F (x) = 1 x+h h f(t)dt (a x b), D = (a, b), H(h) = Nh. x Corollary 3.. Let the conditions of Corollary 3.1 be fulfilled. Then M m b a (3.1) δ n f(x) + b a N, if b a (M m) N (M m)n, if b a (M m) N. Proof. From (3.11) we get δ h f(x) M m b a and if b a (M m)/n we obtain since the function g(y) = M m y y = (M m)/n. + b a N δ h f(x) (M m)n + N y has the minimum (M m)n for

4 60 D.S. MITRINOVIĆ, J.E. PEČARIĆ AND H. KRALJEVIĆ Then Corollary 3.3. Let f : R R be a differentiable function such that m f(x) M, δ h f (x) N (x R, h > 0). (3.13) δ h f(x) (M m)n (x R, h > 0). Proof. Using (.6) (i. e. (3.11) for a = x y, b = x + y), we get (3.14) δ h f(x) M m y + yn. The function g(y) = M m y + yn has the minimum (M m)n for y = (M m)/n, hence for y (M m)/n we get (3.13) from (3.14). Corollary 3.4. Let f : R n R be twice differentiable on D, where D = {x R n ; a i < x i < b i }. Suppose that (3.15) f x i x j N ij on D. Then (3.16) n (b i a i ) f f(a) + f(b) x i 1 N ij [(x i a i )(x j a j ) + (b i x i )(b j x j )] 1 N ij (b i a i )(b j a j ). If, furthermore, (3.17) m f(x) M (x D), then n (b i a i ) f x i (3.18) M m + 1 N ij [(x i a i )(x j a j ) + (b i x i )(b j x j )] M m + 1 N ij (b i a i )(b j a j ).

5 ON LANDAU S THEOREMS 61 Proof. We use Theorem.1 and Remark.3 with X = R n, Y = R, F = f, x = n x i (x R n ), y = y (y R). In this case (3.15) implies F (x) (h, h) = f h i h j x i x j N ij h i h j. So, Theorem.1 implies the first inequalities in (3.16) and (3.18) (note that F (b) F (a) M m). The second inequalities follow from the obvious inequality: ab + cd (a + c)(b + d) (a, b, c, d 0). Corollary 3.5. Let the conditions of Corollary 3.4 be fulfilled and let h = min{b i a i ; 1 i n}. Then { M m (3.19) x i h + h N, if h (M m)/n (M m)n, if h (M m)/n where N = N ij. Proof. We can suppose b i a i = h for every i. Then we get from (3.18) (3.0) x i M m + h h N. Now, as in the proof of Corollary 3., (3.0) implies (3.19). Then (3.1) Corollary 3.6. Let f : R n R be a differentiable function such that m f(x) M and f x i x j N ij on R n. where N = N ij. x i M m N, Proof. For b i = x i + h i, a i = x i h i (h i > 0) (3.18) gives (3.) h i x i M m + 1 N ij h i h j, and for h 1 = = h n = h we obtain (3.3) x i M m + hn h.

6 6 D.S. MITRINOVIĆ, J.E. PEČARIĆ AND H. KRALJEVIĆ Taking the minimum over h > 0 of the right hand side of (3.3) we obtain (3.1). A simple consequence of (3.19) is the following generalization of a result from [4]. Corollary 3.7. Let D = {x R n ; 0 < x i < 1} and let f : D R be a twice differentiable function. Suppose that f(x) 1 (x D) and that (3.15) is fulfilled. Then { N+4 (3.4) x i, if 0 < N 4 N, if N > 4, where N = N ij. Corollary 3.8. Under the assumptions of Corollary 3.6 with (3.5) f x i x j A (x D) instead of (3.15), the following inequality holds true: (3.6) x i (M m)a. i Proof. In the case h 1 = = h n = h we have F(x) (h, h) = h f x i x j h A. Thus, instead of (3.) we obtain (3.7) x i M m + ha h wherefrom (3.6) follows. i Remark 3.9. Corollary 3.8 is a generalization of a result from [17] where the case n = is given. By using (3.6) and (3.7) we easily obtain the following generalization of a result from [15]: Corollary Let f : [0, 1] n R be a twice differentiable function such that f(x) 1 (x [0, 1] n ) and f x i x j A (x (0, 1)n ).

7 ON LANDAU S THEOREMS 63 Then (3.8) x i If f is positive then (3.9) x i i i ( 1,..., 1 ) ( 1,..., 1 ) { + A 4, if 0 < A 8 A, if A 8. { 1 + A 4, if 0 < A 4 A, if A 4. Remark Analogous improvements of Landau s theorems were given by V. M. Olovyanisnikov (see e. g. [16] where some similar results are given). Additional remark. Let us note that results from this paper are given in monograph [13, pp ]. Some further related results are given in [, 3, 5, 6, 7, 10, 14, 8]. References [1] A. Aglic-Aljinović, Lj. Marangunić and J. Pečarić, On Landau type inequalities via extension of Montgomery identity, Euler and Fink identities, Nonlinear functional analysis and applications, to appear. [] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proc. Amer. Math. Soc. 108 (1990), [3] W. Chen and Z. Ditzian, Best approximation and K-functionals, Acta Math. Hungar. 75 (1997), [4] C. K. Chui and P. W. Smith, A note on Landau s problem for bounded intervals, Amer. Math. Monthly 8 (1975), [5] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81 (1998), [6] Z. Ditzian and K. G. Ivanov, Minimal number of significant directional modulii of smoothness, Analysis Math. 19 (1993), [7] Z. Ditzian, Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities, Math. Inequal. Appl. 3 (000), [8] S. S. Dragomir and C. I. Preda, Some Landau type inequalities for functions whose derivatives are Hölder continuous, RGMIA 6 (003), Article 3. [9] R. Ž. Djordjević and G. V. Milovanović, A generalization of E. Landau s theorem, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No (1975), [10] M. K. Kwong and A. Zettl, Norm inequalities for derivatives and differences, Lecture Notes in Mathematics 1536, Springer-Verlag, 199. [11] E. Landau, Einige Ungleichungen für zweimal differentierbare Funktionen, Proc. Lond. Math. Soc. () 13 (1913), [1] Lj. Marangunić and J. Pečarić, On Landau type inequalities for functions with Hölder continuous derivatives, JIPAM. J. Inequal. Pure Appl. Math. 5 (004), Article 7, 5 pp. (electronic). [13] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht/Boston/London, [14] C. P. Niculescu and C. Buse, The Hardy-Landau-Littlewood inequalities with less smoothness, J. Inequal. In Pure and Appl. Math. 4 (003), Article 51.

8 64 D.S. MITRINOVIĆ, J.E. PEČARIĆ AND H. KRALJEVIĆ [15] A. Sharma and J. Tzimbalario, Some inequalities between derivatives on bounded intervals, Delta 6 (1976), [16] S. B. Stečkin, Inequalities between the norms of derivatives for arbitrary functions (Russian), Acta Sci. Math. Szeged 6 (1965), [17] Tcheng Tchou-Yun, Sur les inégalités différentielles, Paris, 1934, 41 pp. J.E. Pečarić Faculty of Technology University of Zagreb Kačićeva ul. 6, Zagreb Croatia pecaric@mahazu.hazu.hr & pecaric@element.hr H. Kraljević Department of Mathematics University of Zagreb Bijenička cesta 30, Zagreb Croatia hrk@math.hr Received: Revised: &