Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES S.S. DRAGOMIR School of Computer Science and Mathematics Victoria University of Technology PO Box 448 Melbourne City MC 800 Victoria, Australia EMail: sever.dragomir@vu.edu.au URL: http://rgmia.vu.edu.au/ssdragomirweb.html volume 4, issue 3, article 63, 003. Received January, 003; accepted 4 May, 003. Communicated by: P.S. Bullen Abstract Home Page c 000 Victoria University ISSN electronic): 443-5756 00-03

Abstract The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with CBS) inequality and provide refinements and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companion, reverse and related results both for real and complex numbers are also presented. 000 Mathematics Subject Classification: 6D5, 6D0. Key words: inequality, Discrete inequalities. Introduction......................................... 6 CBS) Type Inequalities............................ 9. CBS) Inequality for Real Numbers............ 9. CBS) Inequality for Complex Numbers.........3 An Additive Generalisation..................... 3.4 A Related Additive Inequality................... 7.5 A Parameter Additive Inequality................. 9.6 A Generalisation Provided by Young s Inequality....7 Further Generalisations via Young s Inequality..... 4.8 A Generalisation Involving J Convex Functions... 34.9 A Functional Generalisation..................... 36.0 A Generalisation for Power Series................ 40. A Generalisation of Callebaut s Inequality......... 43 Page of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

. Wagner s Inequality for Real Numbers............ 46.3 Wagner s inequality for Complex Numbers........ 49 3 Refinements of the CBS) Inequality.................. 55 3. A Refinement in Terms of Moduli................ 55 3. A Refinement for a Sequence Whose Norm is One.. 59 3.3 A Second Refinement in Terms of Moduli......... 63 3.4 A Refinement for a Sequence Less than the Weights. 67 3.5 A Conditional Inequality Providing a Refinement... 7 3.6 A Refinement for Non-Constant Sequences........ 76 3.7 De Bruijn s Inequality.......................... 83 3.8 McLaughlin s Inequality........................ 84 3.9 A Refinement due to Daykin-Eliezer-Carlitz....... 86 3.0 A Refinement via Dunkl-Williams Inequality...... 89 3. Some Refinements due to Alzer and Zheng........ 9 4 Functional Properties................................. 04 4. A Monotonicity Property....................... 04 4. A Superadditivity Property in Terms of Weights.... 06 4.3 The Superadditivity as an Index Set Mapping...... 09 4.4 Strong Superadditivity in Terms of Weights........ 4.5 Strong Superadditivity as an Index Set Mapping.... 6 4.6 Another Superadditivity Property................ 9 4.7 The Case of Index Set Mapping.................. 4 4.8 Supermultiplicity in Terms of Weights............ 8 4.9 Supermultiplicity as an Index Set Mapping........ 34 5 Reverse Inequalities.................................. 43 5. The Cassels Inequality......................... 43 5. The Pólya-Szegö Inequality..................... 47 Page 3 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

5.3 The Greub-Rheinboldt Inequality................ 49 5.4 A Cassels Type Inequality for Complex Numbers.. 5 5.5 A Reverse Inequality for Real Numbers........... 55 5.6 A Reverse Inequality for Complex Numbers....... 59 5.7 Shisha-Mond Type Inequalities.................. 64 5.8 Zagier Type Inequalities........................ 67 5.9 A Reverse Inequality in Terms of the sup Norm... 7 5.0 A Reverse Inequality in Terms of the Norm..... 75 5. A Reverse Inequality in Terms of the p Norm..... 80 5. A Reverse Inequality Via an Andrica-Badea Result. 84 5.3 A Refinement of Cassels Inequality.............. 88 5.4 Two Reverse Results Via Diaz-Metcalf Results..... 94 5.5 Some Reverse Results Via the Čebyšev Functional.. 98 5.6 Another Reverse Result via a Grüss Type Result.... 08 6 Related Inequalities................................... 7 6. Ostrowski s Inequality for Real Sequences......... 7 6. Ostrowski s Inequality for Complex Sequences..... 9 6.3 Another Ostrowski s Inequality.................. 6.4 Fan and Todd Inequalities....................... 6 6.5 Some Results for Asynchronous Sequences........ 7 6.6 An Inequality via A G H Mean Inequality.... 30 6.7 A Related Result via Jensen s Inequality for Power Functions.................................... 3 6.8 Inequalities Derived from the Double Sums Case... 34 6.9 A Functional Generalisation for Double Sums...... 36 6.0 A CBS) Type Result for Lipschitzian Functions. 39 6. An Inequality via Jensen s Discrete Inequality...... 43 Page 4 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

6. An Inequality via Lah-Ribarić Inequality.......... 44 6.3 An Inequality via Dragomir-Ionescu Inequality..... 47 6.4 An Inequality via a Refinement of Jensen s Inequality50 6.5 Another Refinement via Jensen s Inequality........ 54 6.6 An Inequality via Slater s Result................. 59 6.7 An Inequality via an Andrica-Raşa Result......... 6 6.8 An Inequality via Jensen s Result for Double Sums. 66 6.9 Some Inequalities for the Čebyšev Functional...... 69 6.0 Other Inequalities for the Čebyšev Functional...... 74 6. Bounds for the Čebyšev Functional............... 77 Index............................................... 84 References Page 5 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

. Introduction The inequality, or for short, the CBS) inequality, plays an important role in different branches of Modern Mathematics including Hilbert Spaces Theory, Probability & Statistics, Classical Real and Complex Analysis, Numerical Analysis, Qualitative Theory of Differential Equations and their applications. The main purpose of this survey is to identify and highlight the discrete inequalities that are connected with CBS) inequality and provide refinements and reverse results as well as to study some functional properties of certain mappings that can be naturally associated with this inequality such as superadditivity, supermultiplicity, the strong versions of these and the corresponding monotonicity properties. Many companions and related results both for real and complex numbers are also presented. The first section is devoted to a number of CBS) type inequalities that provides not only natural generalizations but also several extensions for different classes of analytic functions of a real variable. A generalization of the Wagner inequality for complex numbers is obtained. Several results discovered by the author in the late eighties and published in different journals of lesser circulation are also surveyed. The second section contains different refinements of the CBS) inequality including de Bruijn s inequality, McLaughlin s inequality, the Daykin-Eliezer- Carlitz result in the version presented by Mitrinović-Pečarić and Fink as well as the refinements of a particular version obtained by Alzer and Zheng. A number of new results obtained by the author, which are connected with the above ones, are also presented. Page 6 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Section 4 is devoted to the study of functional properties of different mappings naturally associated to the CBS) inequality. Properties such as superadditivity, strong superadditivity, monotonicity and supermultiplicity and the corresponding inequalities are mentioned. In the next section, Section 5, reverse results for the CBS) inequality are surveyed. The results of Cassels, Pólya-Szegö, Greub-Rheinbold, Shisha-Mond and Zagier are presented with their original proofs. New results and versions for complex numbers are also obtained. Reverse results in terms of p norms of the forward difference recently discovered by the author and some refinements of Cassels and Pólya-Szegö results obtained via Andrica-Badea inequality are mentioned. Some new facts derived from Grüss type inequalities are also pointed out. Section 6 is devoted to various inequalities related to the CBS) inequality. The two inequalities obtained by Ostrowski and Fan-Todd results are presented. New inequalities obtained via Jensen type inequality for convex functions are derived, some inequalities for the Čebyşev functionals are pointed out. Versions for complex numbers that generalize Ostrowski results are also emphasised. It was one of the main aims of the survey to provide complete proofs for the results considered. We also note that in most cases only the original references are mentioned. Each section concludes with a list of the references utilized and thus may be read independently. Being self contained, the survey may be used by both postgraduate students and researchers interested in Theory of Inequalities & Applications as well as by Mathematicians and other Scientists dealing with numerical computations, bounds and estimates where the CBS) inequality may be used as a powerful tool. Page 7 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

The author intends to continue this survey with another one devoted to the functional and integral versions of the CBS) inequality. The corresponding results holding in inner-product and normed spaces will be considered as well. Page 8 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

. CBS) Type Inequalities.. CBS) Inequality for Real Numbers The following inequality is known in the literature as Cauchy s or Cauchy- Schwarz s or s inequality. For simplicity, we shall refer to it throughout this work as the CBS) inequality. Theorem.. If ā = a,..., a n ) and b = b,..., b n ) are sequences of real numbers, then ).) a k b k a k b k with equality if and only if the sequences ā and b are proportional, i.e., there is a r R such that a k = rb k for each k {,..., n}. Proof.. Consider the quadratic polynomial P : R R,.) P t) = It is obvious that for any t R. P t) = a k a k t b k ). ) ) t a k b k t + b k Page 9 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Since P t) 0 for any t R it follows that the discriminant of P is negative, i.e., 0 4 = a k b k ) and the inequality.) is proved.. If we use Lagrange s identity ).3) a i b i a i b i = then.) obviously holds. The equality holds in.) iff a k b k a i b j a j b i ) i,j= = a i b j a j b i ) = 0 i<j n a i b j a j b i ) for any i, j {,..., n} which is equivalent with the fact that ā and b are proportional. Remark.. The inequality.) apparently was firstly mentioned in the work [] of A.L. Cauchy in 8. The integral form was obtained in 859 by V.Y. Bunyakovsky []. The corresponding version for inner-product spaces obtained by H.A. Schwartz is mainly known as Schwarz s inequality. For a short history of this inequality see [3]. In what follows we use the spelling adopted in the paper [3]. For other spellings of Bunyakovsky s name, see MathSciNet. Page 0 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.. CBS) Inequality for Complex Numbers The following version of the CBS) inequality for complex numbers holds [4, p. 84]. Theorem.. If ā = a,..., a n ) and b = b,..., b n ) are sequences of complex numbers, then.4) a k b k a k b k, with equality if and only if there is a complex number c C such that a k = c b k for any k {,..., n}. Proof..5). For any complex number λ C one has the equality ak λ b k = = ) ak λ b k āk λb ) k a k + λ b k Re λ If in.5) we choose λ 0 C, n λ 0 := a kb k n b k, b 0 ) a k b k. Page of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

then we get the identity.6) 0 a k λ 0 bk = a k n a kb k n b k, which proves.4). By virtue of.6), we conclude that equality holds in.4) if and only if a k = λ 0 bk for any k {,..., n}.. Using Binet-Cauchy s identity for complex numbers.7) x i y i z i t i = x i t i z i y i x i z j x j z i ) y i t j y j t i ) i,j= = i<j n x i z j x j z i ) y i t j y j t i ) for the choices x i = ā i, z i = b i, y i = a i, t i = b i, i = {,..., n}, we get.8) a i b i a i b i = ā i b j ā j b i i,j= = i<j n ā i b j ā j b i. Page of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Now the inequality.4) is a simple consequence of.8). The case of equality is obvious by the identity.8) as well. Remark.. By the CBS) inequality for real numbers and the generalised triangle inequality for complex numbers z i z i, z i C, i {,..., n} we also have a k b k a k b k ) a k b k. Remark.3. The Lagrange identity for complex numbers stated in [4, p. 85] is wrong. It should be corrected as in.8)..3. An Additive Generalisation The following generalisation of the CBS) inequality was obtained in [5, p. 5]. Theorem.3. If ā = a,..., a n ), b = b,..., b n ), c = c,..., c n ) and d = d,..., d n ) are sequences of real numbers and p = p,..., p n ), q = q,..., q n ) are nonnegative, then.9) p i a i q i b i + p i c i q i d i p i a i c i q i b i d i. Page 3 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

If p and q are sequences of positive numbers, then the equality holds in.9) iff a i b j = c i d j for any i, j {,..., n}. Proof. We will follow the proof from [5]. From the elementary inequality.0) a + b ab for any a, b R with equality iff a = b, we have.) a i b j + c i d j a i c i b j d j for any i, j {,..., n}. Multiplying.) by p i q j 0, i, j {,..., n} and summing over i and j from to n, we deduce.9). If p i, q j > 0 i =,..., n), then the equality holds in.9) iff a i b j = c i d j for any i, j {,..., n}. Remark.4. The condition a i b j = c i d j for c i 0, b j 0 i, j =,..., n) is equivalent with a i c i = d j b j i, j =,..., n), i.e., ā, c and b, d are proportional with the same constant k. Remark.5. If in.9) we choose p i = q i = i =,..., n), c i = b i, and d i = a i i =,..., n), then we recapture the CBS) inequality. The following corollary holds [5, p. 6]. Corollary.4. If ā, b, c and d are nonnegative, then [ ].) a 3 i c i b 3 i d i + c 3 i a i d 3 i b i a i c i b i d i, Page 4 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.3) [ a i b i d i b i a i c i + c i b i d i ] d i a i c i a i b i c i d i ). Another result is embodied in the following corollary [5, p. 6]. Corollary.5. If ā, b, c and d are sequences of positive and real numbers, then: [ ] a 3 i b 3 i.4) + a i c i b i d i a i b i, c i d i.5) [ a i b i c i Finally, we also have [5, p. 6]. b i a i d i + b i c i Corollary.6. If ā, and b are positive, then a 3 ) i b 3 i a i b i b i a i ] ) a i d i a i b i. a i ) b i a i b i 0. Page 5 of 88 The following version for complex numbers also holds. J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem.7. Let ā = a,..., a n ), b = b,..., b n ), c = c,..., c n ) and d = d,..., d n ) be sequences of complex numbers and p = p,..., p n ), q = q,..., q n ) are nonnegative. Then one has the inequality.6) p i a i q i b i + p i c i q i d i The case of equality for p, q positive holds iff a i b j {,..., n}. [ Re p i a i c i Proof. From the elementary inequality for complex numbers with equality iff a = b, we have a + b Re [ a b ], a, b C,.7) a i b j + c i d j Re [ a i c i b j dj ] ] q i b i di. = c i d j for any i, j for any i, j {,..., n}. Multiplying.7) by p i q j 0 and summing over i and j from to n, we deduce.6). The case of equality is obvious and we omit the details. Remark.6. Similar particular cases may be stated but we omit the details. Page 6 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.4. A Related Additive Inequality The following inequality was obtained in [5, Theorem.]. Theorem.8. If ā = a,..., a n ), b = b,..., b n ) are sequences of real numbers and c = c,..., c n ), d = d,..., d n ) are nonnegative, then.8) d i c i a i + c i d i b i c i a i d i b i. If c i and d i i =,..., n) are positive, then equality holds in.8) iff ā = b = k where k = k, k,..., k) is a constant sequence. Proof. We will follow the proof from [5]. From the elementary inequality.9) a + b ab for any a, b R with equality iff a = b; we have.0) a i + b j a i b j for any i, j {,..., n}. Multiplying.0) by c i d j 0, i, j {,..., n} and summing over i from to n and over j from to n, we deduce.8). If c i, d j > 0 i =,..., n), then the equality holds in.8) iff a i = b j for any i, j {,..., n} which is equivalent with the fact that a i = b i = k for any i {,..., n}. The following corollary holds [5, p. 4]. Page 7 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Corollary.9. If ā and b are nonnegative sequences, then [ ].) a 3 i b i + a i b 3 i a i b i ;.) [ a i a i b i + b i ] ) b i a i a i b i. Another corollary that may be obtained is [5, p. 4 5]. Corollary.0. If ā and b are sequences of positive real numbers, then.3).4) and.5) n a i a i + b i a i b i b i + a i + b i a i b i n n a i n a i b i, a i b i b i n, a i b i. The following version for complex numbers also holds. Page 8 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem.. If ā = a,..., a n ), b = b,..., b n ) are sequences of complex numbers, then for p = p,..., p n ) and q = q,..., q n ) two sequences of nonnegative real numbers, one has the inequality [ ].6) q i p i a i + p i q i b i Re p i a i q i bi. For p, q positive sequences, the equality holds in.6) iff ā = b = k = k,..., k). The proof goes in a similar way with the one in Theorem.8 on making use of the following elementary inequality holding for complex numbers.7) a + b Re [ a b ], a, b C; with equality iff a = b..5. A Parameter Additive Inequality The following inequality was obtained in [5, Theorem 4.]. Theorem.. Let ā = a,..., a n ), b = b,..., b n ) be sequences of real numbers and c = c,..., c n ), d = d,..., d n ) be nonnegative. If α, β > 0 and γ R such that γ αβ, then.8) α d i a i c i + β c i b i d i γ c i a i d i b i. Page 9 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. We will follow the proof from [5]. Since α, β > 0 and γ αβ, it follows that for any x, y R one has.9) αx + βy γxy. Choosing in.9) x = a i, y = b j i, j =,..., n), we get.30) αa i + βb j γa i b j for any i, j {,..., n}. If we multiply.30) by c i d j 0 and sum over i and j from to n, we deduce the desired inequality.8). The following corollary holds. Corollary.3. If ā and b are nonnegative sequences and α, β, γ are as in Theorem., then.3) α.3) α b i a i a 3 i + β a i b i + β a i b 3 i γ b i a i b i, ) b i a i γ a i b i. The following particular case is important [5, p. 8]. Page 0 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem.4. Let ā, b be sequences of real numbers. If p is a sequence of nonnegative real numbers with n p i > 0, then: n.33) p i a i p i b i p ia i b n i p n ia i p ib i n p. i In particular,.34) a i b i n a i b i a i b i. Proof. We will follow the proof from [5, p. 8]. If we choose in Theorem., c i = d i = p i i =,..., n) and α = n p ib i, β = n p ia i, γ = n p ia i b i, we observe, by the CBS) inequality with the weights p i i =,..., n) one has γ αβ, and then by.8) we deduce.33). Remark.7. If we assume that ā and b are asynchronous, i.e., a i a j ) b i b j ) 0 for any i, j {,..., n}, then by Čebyšev s inequality.35) p i a i respectively.36) p i b i a i b i n p i p i a i b i a i b i, Page of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

we have the following refinements of the CBS) inequality n p i a i p i b i p ia i b n i p n ia i p ib i.37) n p i ) p i a i b i provided n p ia i b i 0, respectively.38) a i b i a i b i n provided n a ib i 0. a i ) b i a i b i.6. A Generalisation Provided by Young s Inequality The following result was obtained in [5, Theorem 5.]. Theorem.5. Let ā = a,..., a n ), b = b,..., b n ), p = p,..., p n ) and q = q,..., q n ) be sequences of nonnegative real numbers and α, β > with =. Then one has the inequality α + β.39) α q i p i b β i + β p i q i a α i αβ p i b i q i a i. If p and q are sequences of positive real numbers, then the equality holds in.39) iff there exists a constant k 0 such that a α i = b β i = k for each i {,..., n}. Page of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. It is, by the Arithmetic-Geometric inequality [6, p. 5], well known that.40) α x + β y x α y β for x, y 0, α + β =, α, β > with equality iff x = y. Applying.40) for x = a α i, y = b β j i, j =,..., n) we have.4) αb β j + βaα i αβa i b j for any i, j {,..., n} with equality iff a α i = b β j for any i, j {,..., n}. If we multiply.4) by q i p j 0 i, j {,..., n}) and sum over i and j from to n we deduce.39). The case of equality is obvious by the above considerations. The following corollary is a natural consequence of the above theorem. Corollary.6. Let ā, b, α and β be as in Theorem.5. Then.4) α b i a α+ i + β a i b β+ i a i b i ;.43) α a i b i a α i + β b i a i b β i ) a i b i. Page 3 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

The following result which provides a generalisation of the CBS) inequality may be obtained by Theorem.5 as well [5, Theorem 5.]. Theorem.7. Let x and ȳ be sequences of positive real numbers. If α, β are as above, then ).44) x α i y α i + ) x β i α β y β i yi x i y i. The equality holds iff x and ȳ are proportional. Proof. Follows by Theorem.5 on choosing p i = q i = y i, a i = x i y i, b i = x i y i, i {,..., n}. Remark.8. For α = β =, we recapture the CBS) inequality. Remark.9. For a i = z i, b i = w i, with z i, w i C; i =,..., n, we may obtain similar inequalities for complex numbers. We omit the details..7. Further Generalisations via Young s Inequality The following inequality is known in the literature as Young s inequality.45) px q + qy p pqxy, x, y 0 and p + q =, p > with equality iff x q = y p. The following result generalising the CBS) inequality was obtained in [7, Theorem.] see also [8, Theorem ]). Page 4 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem.8. Let x = x,..., x n ), ȳ = y,..., y n ) be sequences of complex numbers and p = p,..., p n ), q = q,..., q n ) be two sequences of nonnegative real numbers. If p >, + =, then p q.46) p p k x k p q k y k p + q q k x k q p k y k q p k x k y k q k x k y k. Proof. We shall follow the proof in [7]. Choosing x = x j y i, y = x i y j, i, j {,..., n}, we get from.45).47) q x i p y j p + p x j q y i q pq x i y i x j y j for any i, j {,..., n}. Multiplying with p i q j 0 and summing over i and j from to n, we deduce the desired result.46). The following corollary is a natural consequence of the above theorem [7, Corollary.] see also [8, p. 05]). Corollary.9. If x and ȳ are as in Theorem.8 and m = m,..., m n ) is a sequence of nonnegative real numbers, then.48) p m k x k p m k y k p + q m k x k q m k y k q Page 5 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

where p >, p + q =. m k x k y k ), Remark.0. If in.48) we assume that m k =, k {,..., n}, then we obtain [7, p. 7] see also [8, p. 05]).49) p x k p y k p + q x k q y k q x k y k ), which, in the particular case p = q = will provide the CBS) inequality. The second generalisation of the CBS) inequality via Young s inequality is incorporated in the following theorem [7, Theorem.4] see also [8, Theorem ]). Theorem.0. Let x, ȳ, p, q and p, q be as in Theorem.8. Then one has the inequality.50) p p k x k p q k y k q + q q k x k q p k y k p p k x k y k p q k x k y k q. Page 6 of 88 Proof. We shall follow the proof in [7]. J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Choosing in.45), x = x j, y = x i y j y i, we get ) q ) p xj xi.5) p + q pq x i x j y j y i y i y j for any y i 0, i, j {,..., n}. It is easy to see that.5) is equivalent to.5) q x i p y j q + p y i p x j q pq x i y i p x j y j q for any i, j {,..., n}. Multiplying.5) by p i q j 0 i, j {,..., n}) and summing over i and j from to n, we deduce the desired inequlality.50). The following corollary holds [7, Corollary.5] see also [8, p. 06]). Corollary.. Let x, ȳ, m and p, q be as in Corollary.9. Then.53) p m k x k p m k y k q + m k x k q m k y k p q m k x k y k p m k x k y k q. Remark.. If in.53) we assume that m k =, k {,..., n}, then we obtain [7, p. 8] see also [8, p. 06]).54) p x k p y k q + q x k q y k p Page 7 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

x k y k p x k y k q, which, in the particular case p = q = will provide the CBS) inequality. The third result is embodied in the following theorem [7, Theorem.7] see also [8, Theorem 3]). Theorem.. Let x, ȳ, p, q and p, q be as in Theorem.8. Then one has the inequality.55) p p k x k p q k y k q + q q k x k p p k y k q p k x k y k p k x k p y k q. Proof. We shall follow the proof in [7]. If we choose x = y i and y = x i y j x j in.45) we get ) q ) p yi xi p + q pq x i y i y j x j x j y j, for any x i, y j 0, i, j {,..., n}, giving.56) q x i p y j q + p y i q x j p pq x i y i x j p y j q for any i, j {,..., n}. Multiplying.56) by p i q j 0 i, j {,..., n}) and summing over i and j from to n, we deduce the desired inequality.55). Page 8 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

The following corollary is a natural consequence of the above theorem [8, p. 06]. Corollary.3. Let x, ȳ, m and p, q be as in Corollary.9. Then one has the inequality:.57) m k x k p m k y k q m k x k y k m k x k p y k q. Remark.. If in.57) we assume that m k =, k = {,..., n}, then we obtain [7, p. 8] see also [8, p. 0]).58) x k p y k q x k y k x k p y k q, which, in the particular case p = q = will provide the CBS) inequality. The fourth generalisation of the CBS) inequality is embodied in the following theorem [7, Theorem.9] see also [8, Theorem 4]). Theorem.4. Let x, ȳ, p, q and p, q be as in Theorem.8. Then one has the inequality.59) q p k x k q k y k q + p p k y k q k x k p q k x k y k p k x k q yk p. Page 9 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. We shall follow the proof in [7]. Choosing in.45), x = x i q yj, y = x j y i p, we get.60) p x i y j q + q x j p y i pq x i q yi p xj y j for any i, j {,..., n}. Multiply.60) by p i q j 0 i, j {,..., n}) and summing over i and j from to n, we deduce the desired inequality.60). The following corollary holds [7, Corollary.0] see also [8, p. 07]). Corollary.5. Let x, ȳ, m and p, q be as in Corollary.9. Then one has the inequality:.6) q m k x k m k y k q + p m k y k m k x k p m k x k y k m k x k q yk p. Remark.3. If in.6) we take m k =, k {,..., n}, then we get.6) q x k y k q + p y k x k p x k y k x k q yk p, which, in the particular case p = q = will provide the CBS) inequality. Page 30 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

The fifth result generalising the CBS) inequality is embodied in the following theorem [7, Theorem.] see also [8, Theorem 5]). Theorem.6. Let x, ȳ, p, q and p, q be as in Theorem.8. Then one has the inequality.63) p p k x k q k y k q + p k y k q k x k p q p k x k p yk q q k x k p y k q. Proof. We will follow the proof in [7]. Choosing in.45), x = y i q, y = x i, y y j x j i, x j 0, i, j {,..., n}, we may write ) y i q ) q x i p p p + q pq y i q xi p, y j x j x j y j from where results.64) p y i x j p + q x i y j q pq x i p yi q xj p y j q for any i, j {,..., n}. Multiplying.64) by p i q j 0 i, j {,..., n}) and summing over i and j from to n, we deduce the desired inequality.63). The following corollary holds [7, Corollary.3] see also [8, p. 08]). p Page 3 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Corollary.7. Let x, ȳ, m and p, q be as in Corollary.9. Then one has the inequality:.65) p m k x k m k y k q + m k y k q m k x k p yk q m k x k p m k x k p y k q. Remark.4. If in.46) we choose m k =, k {,..., n}, then we get [7, p. 0] see also [8, p. 08]).66) p x k y k q + q y k x k p x k p yk q x k p y k q, which in the particular case p = q = will provide the CBS) inequality. Finally, the following result generalising the CBS) inequality holds [7, Theorem.5] see also [8, Theorem 6]). Theorem.8. Let x, ȳ, p, q and p, q be as in Theorem.8. Then one has the inequality:.67) p p k x k q k y k p + q q k y k p k x k q Page 3 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

p k x k p yk q k x k q yk. Proof. We shall follow the proof in [7]. From.45) one has the inequality ).68) q x i p p yj + p x j q q yi ) pq xi p yi x j q yj for any i, j {,..., n}. Multiplying.68) by p i q j 0 i, j {,..., n}) and summing over i and j from to n, we deduce the desired inequality.67). The following corollary also holds [7, Corollary.6] see also [8, p. 08]). Corollary.9. With the assumptions in Corollary.9, one has the inequality.69) m k x k m k p y k p + ) q y k q m k x k p yk m k x k q yk. Remark.5. If in.69) we choose m k = k {,..., n}), then we get.70) x k p y k p + ) q y k q x k p yk x k q yk, which, in the particular case p = q =, provides the CBS) inequality. Page 33 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.8. A Generalisation Involving J Convex Functions For a >, we denote by exp a the function.7) exp a : R 0, ), exp a x) = a x. Definition.. A function f : I R R is said to be J convex on an interval I if ) x + y f x) + f y).7) f for any x, y I. It is obvious that any convex function on I is a J convex function on I, but the converse does not generally hold. The following lemma holds see [7, Lemma 4.3]). Lemma.30. Let f : I R R be a J convex function on I, a > and x, y R\ {0} with log a x, log a y I. Then log a xy I and.73) {exp b [f log a xy )]} exp b [ f loga x )] exp b [ f loga y )] for any b >. Proof. I, being an interval, is a convex set in R and thus log a xy = [ loga x + log a y ] I. Page 34 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Since f is J convex, one has.74) [ f log a xy ) = f loga x + log a y )] f log a x ) + f log a y ). Taking the exp b in both parts, we deduce exp b [f log a xy )] exp b [ f loga x ) + f log a y ) which is equivalent to.73). = { exp b [ f loga x )] exp b [ f loga y )]}, The following generalisation of the CBS) inequality in terms of a J convex function holds [7, Theorem 4.4]. Theorem.3. Let f : I R R be a J convex function on I, a, b > and ā = a,..., a n ), b = b,..., b n ) sequences of nonzero real numbers. If log a a k, log a b k I for all k {,..., n}, then one has the inequality:.75) { exp b [f log a a k b k )] } [ )] n [ exp b f loga a k exp b f loga bk)]. ] Page 35 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. Using Lemma.30 and the CBS) inequality one has exp b [f log a a k b k )] [ [ )] [ expb f loga a k expb f loga bk)]] { [expb [ )]] f loga a k which is clearly equivalent to.75). } ) { [expb [ )]] } f loga b k Remark.6. If in.75) we choose a = b > and f x) = x, x R, then we recapture the CBS) inequality..9. A Functional Generalisation The following result was proved in [0, Theorem ]. Theorem.3. Let A be a subset of real numbers R, f : A R and ā = a,..., a n ), b = b,..., b n ) sequences of real numbers with the properties that i) a i b i, a i, b i A for any i {,..., n}, ii) f a i ), f b i ) 0 for any i {,..., n}, iii) f a i b i ) f a i ) f b i ) for any i {,..., n}. Page 36 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Then one has the inequality: [.76) f a i b i )] f ) n a i f ) b i. Proof. We give here a simpler proof than that found in [0]. We have f a i b i ) f a i b i ) [ )] [ f a i f )] b i [ [f a i )] ) [ = f ) n a i f ) ] b i and the inequality.76) is proved. [f b i )] ) ] by the CBS)-ineq.) Remark.7. It is obvious that for A = R and f x) = x, we recapture the CBS) inequality. Assume that ϕ : N N is Euler s indicator. In 940, T. Popoviciu [] proved the following inequality for ϕ.77) [ϕ ab)] ϕ a ) ϕ b ) for any natural number a, b; Page 37 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

with equality iff a and b have the same prime factors. A simple proof of this fact may be done by using the representation ) ) ϕ n) = n p pk, where n = p α p α p α k k [9, p. 09]. The following generalisation of Popoviciu s result holds [0, Theorem ]. Theorem.33. Let a i, b i N i =,..., n). Then one has the inequality.78) [ ϕ a i b i )] ϕ ) n a i ϕ ) b i. Proof. Follows by Theorem.3 on taking into account that, by.77), [ϕ a i b i )] ϕ a i ) ϕ b i ) for any i {,..., n}. Further, let us denote by s n) the sum of all relatively prime numbers with n and less than n. Then the following result also holds [0, Theorem ]. Theorem.34. Let a i, b i N i =,..., n). Then one has the inequality.79) [ s a i b i )] s ) n a i s ) b i. Page 38 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. It is known see for example [9, p. 09]) that for any n N one has.80) s n) = nϕ n). Thus.8) [s a i b i )] = 4 a i b i ϕ a i b i ) 4 a i b i ϕ a i ) ϕ b i ) = s a i ) s b i ) for each i {,..., n}. Using Theorem.3 we then deduce the desired inequality.79). The following corollaries of Theorem.3 are also natural to be considered [0, p. 6]. Corollary.35. Let a i, b i R i =,..., n) and a >. Denote exp a x = a x, x R. Then one has the inequality.8) [ exp a a i b i )] Corollary.36. Let a i, b i, ) the inequality:.83) [ ] a i b i ) m ) n ) exp a a i exp a b i. i =,..., n) and m > 0. Then one has a i )m b i )m. Page 39 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.0. A Generalisation for Power Series The following result holds [, Remark ]. Theorem.37. Let F : r, r) R, F x) = k=0 α kx k with α k 0, k N. If ā = a,..., a n ), b = b,..., b n ) are sequences of real numbers such that.84) a i b i, a i, b i r, r) for any i {,..., n}, then one has the inequality:.85) F ) n a i F [ ) b i F a i b i )]. Proof. Firstly, let us observe that if x, y R such that xy, x, y r, r), then one has the inequality.86) [F xy)] F x ) F y ). Indeed, by the CBS) inequality, we have.87) [ ] α k x k y k k=0 α k x k k=0 α k y k, n 0. k=0 Taking the limit as n in.87), we deduce.86). Page 40 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Using the CBS) inequality and.86) we have F a i b i ) F a i b i ) [ )] [ F a i F )] b i { [F a i )] ) [ = F ) n a i F ) ] b i, which is clearly equivalent to.85). } [F )] ) b i The following particular inequalities of CBS) type hold [, p. 64].. If ā, b are sequences of real numbers, then one has the inequality.88).89).90) exp ) n a k exp [ ) b k exp a k b k )] ; sinh ) n a k sinh [ ) b k sinh a k b k )] ; cosh ) n a k cosh [ ) b k cosh a k b k )]. Page 4 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

. If ā, b are such that a i, b i, ), i {,..., n}, then one has the inequalities.9).9) tan ) n a k tan [ ) b k tan a k b k )] ; arcsin ) n a k arcsin [ ) b k arcsin a k b k )] ; [ n ) ] [ + a n ) ].93) ln k + b ln k a k b k { [ n ) ]} + ak b k ln ; a k b k [ n ) ] [ n ) ].94) ln ln a k b k { [ n ) ]} ln ; a k b k.95) a k )m [ ] b k )m a k b k ) m, m > 0. Page 4 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.. A Generalisation of Callebaut s Inequality The following result holds see also [, Theorem ] for a generalisation for positive linear functionals). Theorem.38. Let F : r, r) R, F x) = k=0 α kx k with α k 0, k N. If ā = a,..., a n ), b = b,..., b n ) are sequences of nonnegative real numbers such that.96) a i b i, a α i b α i, a α i b α i 0, r) for any i {,..., n} ; α [0, ], then one has the inequality [.97) F a i b i )] F a α i b α i ) n F ) a α i b α i. Proof. Firstly, we note that for any x, y > 0 such that xy, x α y α, x α y α 0, r) one has.98) [F xy)] F x α y α) F x α y α). Indeed, using Callebaut s inequality, i.e., we recall it [4] m ) m m.99) α i x i y i α i x α i y α i α i x α i yi α, we may write, for m 0, that m ).00) α i x i y i i=0 m α i x α y α) i i=0 m α i x α y α) i. i=0 Page 43 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Taking the limit as m, we deduce.98). Using the CBS) inequality and.98) we may write: F a i b i ) F a i b i ) [ )] F a α i b α [ i F )] a α i b α i = { [ [F a α i b α i F a α i b α i which is clearly equivalent to.97). )] ) n ) [F a α i b α i F ) ] a α i b α i The following particular inequalities also hold [, pp. 65-66]. } )] ). Let ā and b be sequences of nonnegative real numbers. Then one has the inequalities.0) [.0) [ exp a k b k )] sinh a k b k )] exp a α k b α k sinh a α k b α k ) n ) n exp a α k b α k ) ; sinh a α k b α k ) ; Page 44 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.03) [ cosh a k b k )] cosh a α k b α k ) n cosh a α k b α k ).. Let ā and b be such that a k, b k 0, ) for any k {,..., n}. Then one has the inequalities: [.04) tan a k b k )] tan ) n a α k b α k tan ) a α k b α k ;.05).06).07) [ arcsin a k b k ) ] arcsin a α k b α k ) n { [ n ) ]} + ak b k ln a k b k [ n + a α ln k b α ) ] [ n k ln a α k b α k { [ n ) ]} ln a k b k arcsin a α k b α k ) ; + a α k b α k a α k b α k ) ] ; Page 45 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

[ n ) ] [ n ) ] ln ln a α k b α k a α. k b α k.. Wagner s Inequality for Real Numbers The following generalisation of the CBS) inequality for sequences of real numbers is known in the literature as Wagner s inequality [5], or [4] see also [4, p. 85]). Theorem.39. Let ā = a,..., a n ) and b = b,..., b n ) be sequences of real numbers. If 0 x, then one has the inequality.08) a k b k + x i j n [ a k + x a i b j ) i<j n a i a j] [ b k + x Proof. We shall follow the proof in [3] see also [4, p. 85]). For any x [0, ], consider the quadratic polynomial in y P y) := x) = x) [ [ a k y b k ) + x a k y b k ) y a k y a k b k + b k ] ] i<j n b i b j ]. Page 46 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

= x) ) + x y a k y +x a k) ) [ a k + x a k y y x) a k ] b k + x) ) = a k + x a k [ y a k b k + x + a k a k ) b k + x b k Since, it is obvious that: ) a k a k b k b k ) + a k b k ) b k + x b k y b k a k = a k b k = b k. i<j n i j n )] a k b k a i a j, a i b j ) b k Page 47 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

and we get ) b k P y) = a k + x i<j n y a k b k + x b k = a i a j ) y i j n i<j n a i b j ) + b i b j, b k + x i<j n b i b j. Taking into consideration, by the definition of P, that P y) 0 for any y R, it follows that the discriminant 0, i.e., 0 4 = a k b k + x a k + x i j n and the inequality.08) is proved. a i b j ) i<j n a i a j) b k + x i<j n b i b j ) Remark.8. If x = 0, then from.08) we recapture the CBS) inequality for real numbers. Page 48 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

.3. Wagner s inequality for Complex Numbers The following inequality which provides a version for complex numbers of Wagner s result holds [6]. Theorem.40. Let ā = a,..., a n ) and b = b,..., b n ) be sequences of complex numbers. Then for any x [0, ] one has the inequality [.09) Re ) a k bk + x Re ) ] a i bj i j n [ a k + x i<j n Re a i ā j ) ] [ b k + x i<j n Proof. Start with the function f : R R,.0) f t) = x) ta k b k + x ta k b k ) We have.) f t) = x) ta k b k ) tā k b ) k + x t a k ) b k t ā k Re b i bj ) ].. ) bk Page 49 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Observe that.) = x) [ t + x = x) [ ] a k t b k ā k t a k bk + b k [ t t + x) a k = a k t a k bk + b k ā k ] b k a k + x a k t Re [ ) a k bk + x Re + x) b k + x b k. = a i ā j i,j= a i + i j n a i ā j a k ]] bk t Page 50 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

and, similarly,.3) Also and thus.4) Re = = b k = a k a i + a i ā j + a i ā j a i + i<j n i<j n j<i n Re a i ā j ) b i + Re ) b i bj. i<j n bk = a i bi + a i bj i j n a k ) bk = Re ) a i bi + Re ) a i bj. i j n Utilising.).4), by.), we deduce [.5) f t) = a k + x ] Re a i ā j ) i<j n [ + Re ) a k bk + x i j n t Re ) ] a i bj t Page 5 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

+ b k + x i<j n Re b i bj ). Since, by.0), f t) 0 for any t R, it follows that the discriminant of the quadratic function given by.5) is negative, i.e., 0 4 [ = Re ) a k bk + x Re ) ] a i bj i j n [ a k + x ] [ Re a i ā j ) b k + x Re ) ] b i bj i<j n i<j n and the inequality.09) is proved. Remark.9. If x = 0, then we get the CBS) inequality [.6) Re ) ] a k bk a k b k. Page 5 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

References [] V.Y. BUNIAKOWSKI, Sur quelques inégalités concernant les intégrales aux differences finies, Mem. Acad. St. Petersburg, 7) 859), No. 9, -8. [] A.L. CAUCHY, Cours d Analyse de l École Royale Polytechnique, I re Partie, Analyse Algébrique, Paris, 8. [3] P. SCHREIDER, The inequality, Hermann Graßmann Lieschow, 994), 64 70. [4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 993. [5] S.S. DRAGOMIR, On some inequalities Romanian), Caiete Metodico Ştiinţifice, No. 3, 984, pp. 0. Faculty of Mathematics, Timişoara University, Romania. [6] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin-Göttingen-Heidelberg, 96. [7] S.S. DRAGOMIR, On Cauchy-Buniakowski-Schwartz s Inequality for Real Numbers Romanian), Caiete Metodico-Ştiinţifice, No. 57, pp. 4, 989. Faculty of Mathematics, Timişoara University, Romania. [8] S.S. DRAGOMIR AND J. SÁNDOR, Some generalisations of Cauchy- Buniakowski-Schwartz s inequality Romanian), Gaz. Mat. Metod. Bucharest), 990), 04 09. Page 53 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

[9] I. CUCUREZEANU, Problems on Number Theory Romanian), Ed. Technicǎ, Bucharest, 976. [0] S.S. DRAGOMIR, On an inequality of Tiberiu Popoviciu Romanian), Gaz. Mat. Metod., Bucharest) 8 987), 4 8. ZBL No. 7:0A. [] T. POPOVICIU, Gazeta Matematicǎ, 6 940), p. 334. [] S.S. DRAGOMIR, Inequalities of Cauchy-Buniakowski-Schwartz s type for positive linear functionals Romanian), Gaz. Mat. Metod. Bucharest), 9 988), 6 64. [3] T. ANDRESCU, D. ANDRICA AND M.O. DRÎMBE, The trinomial principle in obtaining inequalities Romanian), Gaz. Mat. Bucharest), 90 985), 33 338. [4] P. FLOR, Über eine Unglichung von S.S. Wagner, Elemente Math., 0 965), 36. [5] S.S. WAGNER, Amer. Math. Soc., Notices, 965), 0. [6] S.S. DRAGOMIR, A version of Wagner s inequality for complex numbers, submitted. Page 54 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

3. Refinements of the CBS) Inequality 3.. A Refinement in Terms of Moduli The following result was proved in []. Theorem 3.. Let ā = a,..., a n ) and b = b,..., b n ) be sequences of real numbers. Then one has the inequality 3.) a k ) b k a k b k a k a k b k b k a k b k Proof. We will follow the proof from []. For any i, j {,..., n} the next elementary inequality is true: 3.) a i b j a j b i a i b j a j b i. By multiplying this inequality with a i b j a j b i 0 we get a k b k 0. 3.3) a i b j a j b i ) a i b j a j b i ) a i b j a j b i ) = a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i. Page 55 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Summing 3.3) over i and j from to n, we deduce a i b j a j b i ) i,j= a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i i,j= a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i ), i,j= giving the desired inequality 3.). The following corollary is a natural consequence of 3.) [, Corollary 4]. Corollary 3.. Let ā be a sequence of real numbers. Then 3.4) n a k n ) a k n a k a k n a k n a k 0. There are some particular inequalities that may also be deduced from the above Theorem 3. see [, p. 80]).. Suppose that for ā and b sequences of real numbers, one has sgn a k ) = Page 56 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

sgn b k ) = e k {, }. Then one has the inequality 3.5) a k ) b k a k b k e k a k ) e k b k e k a k b k 0.. If ā = a,..., a n ), then we have the inequality 3.6) n a k [ ] ) k a k a k 3. If ā = a,..., a n+ ), then we have the inequality n+ 3.7) n + ) a k n+ ) k a k ) ) k a k 0. n+ n+ a k ) k a k 0. The following version for complex numbers is valid as well. Page 57 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem 3.3. Let ā = a,..., a n ) and b = b,..., b n ) be sequences of complex numbers. Then one has the inequality 3.8) a i b i a i b i a i ā i b i b i Proof. We have for any i, j {,..., n} that a i b i b i ā i 0. ā i b j ā j b i a i b j a j b i. Multiplying by ā i b j ā j b i 0, we get ā i b j ā j b i a i ā i b j b j + a j ā j b i b i a i b i b j ā j b i ā i a j b j. Summing over i and j from to n and using the Lagrange s identity for complex numbers: a i b i a i b i = ā i b j ā j b i we deduce the desired inequality 3.8). i,j= Remark 3.. Similar particular inequalities may be stated, but we omit the details. Page 58 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

3.. A Refinement for a Sequence Whose Norm is One The following result holds [, Theorem 6]. Theorem 3.4. Let ā = a,..., a n ), b = b,..., b n ) be sequences of real numbers and ē = e,..., e n ) be such that n e i =. Then the following inequality holds 3.9) a i b i [ a k b k e k a k ) a k b k. Proof. We will follow the proof from []. From the CBS) inequality, one has 3.0) [ ) ] a k e i a i e k e k b k + e k a k [ ) ] b k e i b i e k ] e k b k { [ ) ] ) ]} a k e i a i e k [b k e i b i e k. Page 59 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Since n e k =, a simple calculation shows that [ ) ] ) a k e i a i e k = a k e k a k, [ ) ] b k e i b i e k = ) b k e k b k, and [ ) ] ) ] a k e i a i e k [b k e i b i e k = a k b k and then the inequality 3.0) becomes ) ) 3.) a k e k a k b k e k b k a k b k e k a k e k a k Using the elementary inequality m l ) p q ) mp lq), m, l, p, q R e k b k ) e k b k 0. Page 60 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

for the choices ) m = a k, l = e k a k, p = and q = e k b k b k ) the above inequality 3.) provides the following result 3.) Since a k ) a k ) b k ) e k a k e k b k a k b k e k a k b k ) e k a k e k b k e k b k. then, by taking the square root in 3.) we deduce the first part of 3.9). The second part is obvious, and the theorem is proved. The following corollary is a natural consequence of the above theorem [, Corollary 7]. Page 6 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Corollary 3.5. Let ā, b, ē be as in Theorem 3.4. If n a kb k = 0, then one has the inequality: 3.3) a k ) ) b k 4 e k a k e k b k. The following inequalities are interesting as well [, p. 8].. For any ā, b one has the inequality 3.4) a k b k [ a k b k n ) a k b k.. If n a kb k = 0, then 3.5) a k a k b k + n ) b k 4 ) a n k b k. a k ] b k In a similar manner, we may state and prove the following result for complex numbers. Page 6 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem 3.6. Let ā = a,..., a n ), b = b,..., b n ) be sequences of complex numbers and ē = e,..., e n ) a sequence of complex numbers satisfying the condition n e i =. Then the following refinement of the CBS) inequality holds 3.6) a i b i [ a k bk a k bk. a k ē k e k bk + a k ē k ] e k bk The proof is similar to the one in Theorem 3.4 on using the corresponding CBS) inequality for complex numbers. Remark 3.. Similar particular inequalities may be stated, but we omit the details. 3.3. A Second Refinement in Terms of Moduli The following lemma holds. Lemma 3.7. Let ā = a,..., a n ) be a sequence of real numbers and p = p,..., p n ) a sequence of positive real numbers with n p i =. Then one Page 63 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

has the inequality: ) 3.7) p i a i p i a i p i a i a i p i a i p i a i. Proof. By the properties of moduli we have a i a j ) = a i a j ) a i a j ) a i a j ) a i a j ) for any i, j {,..., n}. This is equivalent to 3.8) a i a i a j + a j a i a i + a j a j a i a j a j a i for any i, j {,..., n}. If we multiply 3.8) by p i p j 0 and sum over i and j from to n we deduce p j j= p i a i p i a i p j a j + p i j= j= p j a j p i p ai j a i + a j a j a i a j a j a i i,j= p i p j a i a i + a j a j a i a j a j a i ), i,j= which is clearly equivalent to 3.7). Page 64 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Using the above lemma, we may prove the following refinement of the CBS) -inequality. Theorem 3.8. Let ā = a,..., a n ) and b = b,..., b n ) be two sequences of real numbers. Then one has the inequality ) 3.9) b i a i b i a i a i sgn a i ) b i b i a i b i Proof. If we choose for a i 0, i {,..., n}) in 3.7), that we get a i n a k from where we get b i n a k p i := bi a i a i n, x i = b i, i {,..., n}, a k a i ) a i n a k b i a i n a ib i ) n a k ) a i n a k bi a i bi a i n a i a i n a k ) a i n a k b i a i a i b i 0. a i n a k bi a i b i b n i a ib i n a ib i n a k ) Page 65 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

which is clearly equivalent to 3.9). The case for complex numbers is as follows. Lemma 3.9. Let z = z,..., z n ) be a sequence of complex numbers and p = p,..., p n ) a sequence of positive real numbers with n p i =. Then one has the inequality: 3.0) p i z i p i z i p i z i z i p i z i Proof. By the properties of moduli for complex numbers we have z i z j z i z j ) z i z j ) for any i, j {,..., n}, which is clearly equivalent to p i z i. z i Re z i z j ) + z j z i z i + z j z j z i z j z i z j for any i, j {,..., n}. If we multiply with p i p j 0 and sum over i and j from to n, we deduce the desired inequality 3.0). Now, in a similar manner to the one in Theorem 3.8, we may state the following result for complex numbers. Page 66 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Theorem 3.0. Let ā = a,..., a n ) a i 0, i =,..., n) and b = b,..., b n ) be two sequences of complex numbers. Then one has the inequality: 3.) a i b i ā i b i a i b i b i a i b i ā i b i 0. a i 3.4. A Refinement for a Sequence Less than the Weights The following result was obtained in [, Theorem 9] see also [, Theorem 3.0]). Theorem 3.. Let ā = a,..., a n ), b = b,..., b n ) be sequences of real numbers and p = p,..., p n ), q = q,..., q n ) be sequences of nonnegative real numbers such that p k q k for any k {,..., n}. Then we have the inequality 3.) p k a k p k b k ) p k q k ) a k b k + q k a k q k b k [ ] ) p k q k ) a k b k + q k a k b k p k a k b k. Page 67 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003

Proof. We shall follow the proof in []. Since p k q k 0, then the CBS) inequality for the weights r k := p k q k will give 3.3) p k a k ) q k a k p k b k Using the elementary inequality for the choices and ac bd) a b ) c d ), ) a = p k a k, b = q k a k d = q k b k ) q k b k [ ] p k q k ) a k b k. a, b, c, d R ), c = ) p k b k ) Page 68 of 88 J. Ineq. Pure and Appl. Math. 43) Art. 63, 003