Hindawi Publishing Corporation Journal ofInequalitiesand Applications Volume 2008, Article ID 890137, 7 pages doi:10.1155/2008/890137 Research ArticleSomeMultiplicativeInequalitiesforInnerProductsandoftheCarlson Type Sorina Barza, 1 Lars-Erik Persson, 2 and Emil C. Popa 3 1 Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden 2 Department of Mathematics, Lule ˚ a University of Technology, 97187 Lule ˚ a, Sweden 3 Department of Mathematics, “Lucian Blaga” University of Sibiu, 550024 Sibiu, Romania Correspondence should be addressed to Sorina Barza, sorina.barza@kau.se Received 26 September 2007; Accepted 17 January 2008 Recommended by Wing-Sum Cheung We prove a multiplicative inequality forinner products, which enables us to deduce improvements ofinequalitiesoftheCarlson type for complex functions and sequences, and also other known inequalities. Copyright q 2008 Sorina Barza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let a n ∞ n1 be a nonzero sequence of nonnegative numbers and let f be a measurable function on 0, ∞. In 1934, Carlson 1 proved that theinequalities ∞ n1 a n 4 <π 2 ∞ n1 a 2 n ∞ n1 n 2 a 2 n , 1.1 ∞ 0 fxdx 4 ≤ π 2 ∞ 0 f 2 xdx ∞ 0 x 2 f 2 xdx 1.2 hold and C π 2 is the best constant in both cases. Several generalizations and applications in different branches of mathematics were given during the years. For a complete survey ofthe results and applications concerning the above inequalitiesand also historical remarks, see the book 2. In particular, somemultiplicativeinequalitiesofthe type ∞ 0 fxdx 4 ≤ C ∞ 0 w 2 1 xf 2 xdx ∞ 0 w 2 2 xf 2 xdx 1.3 2 Journal ofInequalitiesand Applications are known for special weight functions w 1 and w 2 , where usually w 1 and w 2 are power func- tions or homogeneous. In this paper, we prove a refined version of 1.3 for a fairly general class of weight functions see Corollary 3.2. In particular, this inequality shows that 1.2 holds with the constant π 2 for many infinite weights beside the classical ones w 1 x1andw 2 xx. Our method of proof is different from the other proofs e.g., those by Larsson et al. presented in the book 2 andthe basic idea is to first prove a more general multiplicative inequality forinnerproducts see Theorem 2.3. Some similar improvements and complements of 1.1 are also included. The paper is organized as follows: in Section 2 we prove our general multiplicative in- equality forinner products. In Section 3 we deduce an integral inequality oftheCarlson type for general measure spaces and prove some corollaries forthe Lebesgue measure andthe counting measure, which are improvements ofinequalities 1.2 and 1.1. Section 4 is devoted to an inequality for an inner product defined on a space of square matrices, which is a general- ization of known discrete inequalities. 2. A multiplicative inequality forinnerproducts Let X, , · be a vector space over a scalar field R or C and let F : X × X → C be an inner product on X. First, we formulate the following Lemma. Lemma 2.1. Let x, y ∈ X be such that x, y / 0. Then there exists λ ∈ C, λ / 0 such that Reλ/ λFx, y 0 and |λ| 2 Fy,y/Fx, x. Proof. Let x, y ∈ X be such that x, y / 0. Then Fx, x,Fy, y > 0, and Fx, y|Fx, y|e iϕ forsome ϕ ∈ 0, 2π. If Fx, y0, then ϕ is arbitrary. Set λ 4 Fy, y/Fx, xe i±π/4−ϕ/2 . Then |λ| 2 Fy, y/Fx, x > 0, λ/λ e i±π/2−ϕ ,andλ/λFx, y|Fx, y|e ±iπ/2 ,so Reλ/ λFx, y 0 andthe proof is completed. Remark 2.2. It is observed that the same result c an be achieved also with λ 4 Fy, y/Fx, xe i±3π/4−ϕ/2 . Thus, for Fx, y ∈ R,wehaveλ p ± pi,wherep 2 1/2 Fy, y/Fx, x. Our multiplicative inequality oftheCarlson type reads as follows. Theorem 2.3. Let x, y, v ∈ X be such that x, y / 0 and let λ be any ofthe numbers satisfying the conditions of Lemma 2.1. Then the inequality F λx 1 λ y, v 4 ≤ 4Fx, xFy,yF 2 v, v2.1 holds. Proof. By using Schwarz inequality, we find that F λx 1 λ y, v 2 ≤ F λx 1 λ y, λx 1 λ y Fv, v |λ| 2 Fx, x 1 |λ| 2 Fy, y2Re λ λ Fx, y Fv, v. 2.2 Sorina Barza et al. 3 By, now, applying Lemma 2.1 and our assumptions on λ, we find that the right-hand side of 2.2 is equal to 2 Fx, x Fy, yFv, v and 2.1 follows. 3. InequalitiesoftheCarlson type Let Ω,dμ be a measure space and let f, g : Ω → C be measurable functions. We define Ff, g Ω ftgtdμ 3.1 which is a standard inner product on L 2 Ω,dμ. Now, we state and prove the following new Carlson-type inequality. Theorem 3.1. Let f : Ω → C and w 1 ,w 2 : Ω → R be such that w 1 f, w 2 f / 0 a.e.,f ∈ L 2 w 2 1 Ω,dμ L 2 w 2 2 Ω,dμ and |λ| 2 w 2 1 1/|λ| 2 w 2 2 > 0,where λ p ± pi, p 2 1/2 Ω w 2 2 xfxdμ Ω w 2 1 xfxdμ . 3.2 Then Ω fxdμ 4 ≤4 Ω dμ λw 1 x1/λw 2 x 2 2 Ω w 2 1 x fx 2 dμ Ω w 2 2 x fx 2 dμ . 3.3 Proof. In theinner product defined in 3.1 we substitute fw 1 and fw 2 for respectively f and g and observe that in this case the number Ffw 1 ,fw 2 Ω w 1 xw 2 x|fx| 2 dμ is real. Since Im Fw 1 f, w 2 f0, by arguing as in the proof of Lemma 2.1 we find that λ p ± pi,where p 2 1/2 Ω w 2 2 x|fx| 2 dμ/ Ω w 2 1 x|fx| 2 dμ fulfills the conditions of Theorem 2.3,sothe inequality 3.3 follows from the inequality 2.1 by taking vx1/λw 1 x1/λw 2 x. The proof is complete. The following corollary ofthe above theorem is an improvement of 3, Theorem 2.1. Corollary 3.2. For a ∈ R,letf : a, ∞ → C be an integrable function and let w 1 ,w 2 : a, ∞ → R be two continuously di fferentiable functions such that 0 <m inf x>a w 2 xw 1 x −w 2 xw 1 x < ∞ and lim x→∞ w 2 x/w 1 x∞.Then ∞ a fxdx 4 ≤ ⎛ ⎜ ⎝ π m − 2 m arctan w 2 a ∞ a w 2 1 x fx 2 dx w 1 a ∞ a w 2 2 x fx 2 dx ⎞ ⎟ ⎠ 2 × ∞ a w 2 1 x fx 2 dx ∞ a w 2 2 x fx 2 dx . 3.4 4 Journal ofInequalitiesand Applications Remark 3.3. Forthe special case when a 0, w 2 00, and m 1, the inequality 3.4 reads ∞ 0 fxdx 4 ≤ π 2 ∞ 0 w 2 1 x fx 2 dx ∞ 0 w 2 2 x fx 2 dx 3.5 and also this generalization of 1.2 seems to be new see 2 andthe references given there. Proof. Let Ωa, ∞ and μ be the Lebesgue measure in inequality 3.3 . Easy calculations show that our assumptions imply that λ 2 w 2 x w 1 x 2 |λ| 4 w 2 2 x w 2 1 x , 1 m w 2 x w 1 x ≥ 1 w 2 1 x . 3.6 Hence, we get that ∞ a dx λw 1 x1/λw 2 x 2 ∞ a |λ| 2 /w 2 1 x λ 2 w 2 x/w 1 x 2 dx ∞ a 1/|λ| 2 w 2 1 x 1 w 2 x/|λ| 2 w 1 x 2 dx ≤ 1 m ∞ a w 2 x/|λ| 2 w 1 x 1 w 2 x/|λ| 2 w 1 x 2 dx 1 m arctan w 2 x |λ| 2 w 1 x ∞ a π 2m − 1 m arctan w 2 a ∞ a w 2 1 x fx 2 dx w 1 a ∞ a w 2 2 x fx 2 dx 3.7 and, by using Theorem 3.1, the proof follows. Remark 3.4. As in 4, we can prove that the condition lim x→∞ w 2 x/w 1 x∞ cannot be weakened, it is also necessary for our inequality. Let, now, Ω N and X l 2 w a a n ∞ n1 : a n ∈ C, ∞ n1 a n 2 w n < ∞ , 3.8 where w w n ∞ n1 is a nontrivial sequence of nonnegative real numbers. Then the functional Fa, b ∞ n1 a n b n w n 3.9 is obviously an inner product on l 2 w. Now, we are able to state the following result which is a direct consequence of Theorem 3.1. Sorina Barza et al. 5 Corollary 3.5. Let α n ∞ n1 , β n ∞ n1 be two nontrivial sequences of complex numbers. Then ∞ n1 a n w n 4 ≤ 4 ∞ n1 w n λα n 1/λβ n 2 2 ∞ n1 |α| 2 n a n 2 w n ∞ n1 |β| 2 n a n 2 w n 3.10 for any sequence a n ∞ n1 ⊂ C of complex numbers, where λ p ± pi, p 2 1/2 ∞ n1 α 2 n |a n | 2 w n ∞ n1 β 2 n |a n | 2 w n . 3.11 Proof. The proof follows by using Theorem 3.1 with Ω N and dμ ∞ i1 w i δ i . Finally, we also include another discrete Carlson-type inequality for complex sequences, which in particular generalizes 3, Theorem 3.1. Corollary 3.6. Let a n ∞ n1 be a sequence of complex numbers and let αx,βx be two positive con- tinuously differentiable functions on 0, ∞ such that 0 <m inf x>0 β xαx − βxα x < ∞. Suppose also that αx is increasing, lim x→∞ βx/αx∞ and lim x→0 βx/αx0. Then the following inequality holds: ∞ n1 a n 4 ≤ π m − 2|λ| 2 ∞ n1 |λ| 4 α c n α c n β c n β c n |λ| 4 α 2 c n β 2 c n 2 2 × ∞ n1 a n 2 α 2 n ∞ n1 a n 2 β 2 n , 3.12 forsome numbers c n ∈ n − 1,n, n ∈ N,whereλ ∈ C is such that |λ| 2 ∞ n1 β 2 n a n 2 ∞ n1 α 2 n a n 2 . 3.13 Remark 3.7. Forthe special case when m 1 i.e., when inf x>0 β xαx1, the inequality ∞ n1 a n 4 ≤ π 2 ∞ n1 a n 2 α 2 n ∞ n1 a n 2 β 2 n 3.14 and also the generalization of inequality 1.1 in this simple form seem to be new. Proof. Let w n 1 for any n ∈ N, α n αn and β n βn in Corollary 3.5.Wehavealso ∞ n1 1 λα n 1/λβ n 2 ∞ n1 |λ| 2 /α 2 n |λ| 4 β 2 n /α 2 n . 3.15 6 Journal ofInequalitiesand Applications Fix N ∈ N. Since the function ϕx|λ| 2 /αx 2 /|λ| 4 β 2 x/α 2 x is decreasing, w e have that ∞ n1 |λ| 2 /α 2 n |λ| 4 β 2 n /α 2 n < ∞ N ϕxdx N n1 ϕn ∞ 0 ϕxdx − N 0 ϕxdx − N n1 ϕn ≤ 1 m arctan βx |λ| 2 αx ∞ 0 − N n1 n n−1 ϕx − ϕn dx ≤ π 2m 1 2 N n1 ϕ c n , 3.16 where c n are points between n − 1andn from the Lagrange mean-value theorem. By differen- tiating, we find that ∞ n1 1 λα n 1/λβ n 2 ≤ π 2m −|λ| 2 N n1 |λ| 4 αc n α c n βc n β c n |λ| 4 α 2 c n β 2 c n 2 , 3.17 where |λ| 2 β 2 n a n 2 α 2 n a n 2 3.18 which, by letting N →∞and using 3.10, implies 3.12, andthe proof is complete. 4. Multiplicativeinequalitiesfor matrices Let n ∈ N and X be the vector space of n × n complex matrices. We denote by trA the trace ofthe matrix A and by A ∗ the Hermitian adjoint of A,thatis,A ∗ A t .Itiswellknownthat AB ∗ B ∗ A ∗ and A ∗ ∗ A; see, for example, 5. Moreover, a matrix A is called unitary if AA ∗ I n ,whereI n is the unity matrix see, e.g., 5. We define FA, Btr B ∗ A 4.1 which is an inner product on X since FA B, CtrC ∗ A B trC ∗ AtrC ∗ B FA, CFB, C.Wehavealsothat FA, Btr B ∗ A n j1 n k1 a kj b kj n j1 n k1 a kj b kj tr A ∗ B . 4.2 The other properties oftheinner product are obvious. The inequality 2.1 becomes in this case tr C ∗ λA 1 λ B 4 ≤ 4tr 2 C ∗ C tr A ∗ A tr B ∗ B , 4.3 where λ is one ofthe complex numbers satisfying the conditions of Lemma 2.1.Wecannow formulate the following result. Sorina Barza et al. 7 Proposition 4.1. Let P, W 1 ,W 2 be n × n complex matrices such that W 1 P, W 2 P / 0.Then trP 4 ≤ 4tr 2 λW 1 1 λ W 2 −1 λW 1 1 λ W 2 −1 ∗ tr P ∗ W ∗ 1 W 1 P tr P ∗ W ∗ 2 W 2 P , 4.4 where λ ∈ C is the parameter defined in Lemma 2.1 (related to the matrices W 1 P and W 2 P), such that λW 1 1/λW 2 is a regular matrix. Proof. If we substitute A W 1 P, B W 2 P, C λW 1 1/λW 2 −1 ∗ in 4.3, we get inequality 4.4. Remark 4.2. If W 1 W 2 √ 2/2W where W is a unitary matrix, then λ √ 2/2 √ 2/2i satisfies the conditions of Lemma 2.1. Since λ 1/λ √ 2, the inequality 4.4 becomes trP 2 ≤ ntr P ∗ P 4.5 and it holds for any n × n complex matrix P. In particular, for diagonal matrices P diaga 1 , ,a n , we get the well-known inequality n k1 a k 2 ≤ n n k1 a k 2 4.6 for a k ∈ C, k 1, ,n. Acknowledgments The authors thank the referees forsome valuable comments and remarks. They also thank one ofthe referees forthe generosity to even suggest simplifications of one ofthe proofs. References 1 F. Carlson, “Une in ´ egalit ´ e,” Arkiv f ¨ or Matematik, Astronomi och Fysik B, vol. 25, no. 1, pp. 1–5, 1934. 2 L. Larsson, L. Maligranda, J. Pe ˇ cari ´ c, and L E. Persson, MultiplicativeInequalitiesofCarlson Type and Interpolation, World Scientific, Hackensack, NJ, USA, 2006. 3 S. Barza and E. C. Popa, “Weighted multiplicative integral inequalities,” Journal ofInequalities in Pure and Applied Mathematics, vol. 7, no. 5, article 169, p. 6, 2006. 4 L. Larsson, “A new Carlson type inequality,” Mathematical Inequalities & Applications, vol. 6, no. 1, pp. 55–79, 2003. 5 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 890137, 7 pages doi:10.1155/2008/890137 Research Article Some Multiplicative Inequalities for Inner Products and of the Carlson. prove a multiplicative inequality for inner products, which enables us to deduce improvements of inequalities of the Carlson type for complex functions and sequences, and also other known inequalities. Copyright. thank the referees for some valuable comments and remarks. They also thank one of the referees for the generosity to even suggest simplifications of one of the proofs. References 1 F. Carlson,