ON A SINE POLYNOMIAL OF TURÁN

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number 1, 18 ON A SINE POLYNOMIAL OF TURÁN HORST ALZER AND MAN KAM KWONG ABSTRACT. In 1935, Turán proved that ( n + a j ) S n,a() = sin(j) >, n j n, a N, < < π. We present various related inequalities. Among others, we show that the refinements S n 1,a () sin() and S n,a () sin()(1 + cos()) are valid for all integers n 1 and real numbers a 1 and (, π). Moreover, we apply our theorems on sine sums to obtain inequalities for Chebyshev polynomials of the second kind. 1. Introduction. The sequence σ n,k (z) is recursively defined by σ n, (z) = z j, σ n,k (z) = σ j,k 1 (z), k N. Then, we have σ n,k (e i ) = j= j= j= ( ) n + k j cos(j) + i k ( n + k j k ) sin(j). In 1935, Turán [1] studied the imaginary part and proved by induction on n and k the remarkable inequality ( ) n + k j (1.1) sin(j) >, n, k N, < < π. k 1 AMS Mathematics subject classification. Primary 6D5, 6D15, 33C45. Keywords and phrases. Trigonometric sums, inequalities, Chebyshev polynomials. Research of the second author was supported by the Hong Kong government, GRF grant No. PolyU 53/1P and the Hong Kong Polytechnic University, grant Nos. G-UC and G-UA1. Received by the editors on June 13, 16, and in revised form on October 14, 16. DOI:1.116/RMJ Copyright c 18 Rocky Mountain Mathematics Consortium 1

2 HORST ALZER AND MAN KAM KWONG In the same paper, Turán presented an elegant inequality for a sine sum in two variables. He demonstrated that the following companion of (1.1) is valid: ( ) n + k j sin(j) sin(jy) (1.) >, n, k N, <, y < π. k j Szegő [11] applied (1.1) with k = to establish a theorem on univalent functions. An etension of (1.1) with k = was given by Alzer and Kwong [3], whereas Alzer and Fuglede [1] offered a positive lower bound for the sine polynomial in (1.1) under the assumption that n, k. In fact, they proved that (1.3) ( n + k j k ) sin(j) > (π ), n, k N, < < π. π The definition of the sums given in (1.1) and (1.) requires that k be a nonnegative integer. However, the identity ( ) ( ) n + k j n + k j = k n j reveals that, if we use the second binomial coefficient, then k can be any real number. Therefore, it is natural to ask for all real parameters a and b such that we have, for all integers n 1 and real numbers (, π), y (, π), ( ) n + a j S n,a () = sin(j) > n j and (1.4) Θ n,b (, y) = ( ) n + b j sin(j) sin(jy) >. n j j In this paper, we solve both problems. Moreover, we provide several closely related inequalities. Among others, we show that there eist functions λ(), µ() and λ (, y), µ (, y) such that the estimates S n 1,a () λ() >, S n,a () µ() >

3 ON A SINE POLYNOMIAL OF TURÁN 3 and Θ n 1,a (, y) λ (, y) >, Θ n,a (, y) µ (, y) > are valid for all integers n 1 and real numbers a 1, (, π), y (, π). In the net section, we collect some lemmas which are needed to prove our main results given in Section 3. Finally, in Section 4, we apply our theorems to obtain inequalities for sums involving Chebyshev polynomials of the second kind, and we also offer new integral inequalities for these polynomials. Throughout, we maintain the notation introduced in this section. For more information on inequalities for trigonometric sums and polynomials, the interested reader is referred to the monograph by Milovanović, Mitrinović, Rassias [9, Chapter 6].. Lemmas. The first lemma is due to Fejér [5]. Lemma.1. Let (, π), and let n 1 (.1) ϕ n () = sin(j) + sin(n). Then, ϕ n () > for n = 1, and ϕ n () for n 3. In order to prove (.1), Fejér made use of the identity ( n 1 j ) ϕ n () = sin() n + cos(k) k=1 which he obtained by comparing the coefficients of certain power series. Here, we offer a different proof which is more elementary than Fejér s approach. Proof. Multiplying both sides of (.1) by sin(/) gives n 1 sin(/)ϕ n () = sin(/) sin(j) + sin(/) sin(n) n 1 = (cos((j 1/)) cos((j + 1/)))

4 4 HORST ALZER AND MAN KAM KWONG + 1 (cos((n 1/)) cos((n + 1/))) = cos(/) 1 cos((n 1/)) 1 cos((n + 1/)) = cos(/) cos(/) cos(n) = cos(/) ( 1 cos(n) ). Since (, π), we conclude that ϕ n () > for n = 1, and ϕ n () for n 3. Net, an identity is presented which will be a helpful tool not only to establish our inequalities for trigonometric sums but also to cover all cases of equality. Lemma.. Let c k, k = 1,..., n, be real numbers and Then, (.) n k γ k,n = c k + ( 1) j c j+k, k = 1,..., n. c j sin(j) = where ϕ j () is defined in (.1). γ j,n ϕ j (), Proof. We have ( n j ) γ j,n ϕ j () = c j ϕ j () + ϕ j () ( 1) k c k+j = c j ϕ j () + = c 1 ϕ 1 () + = c 1 sin() + = k=1 j 1 (c j ) ( 1) j k ϕ k () j= k=1 j 1 ) c j (ϕ j () + ( 1) j k ϕ k () j= c j sin(j) j= c j sin(j). k=1

5 ON A SINE POLYNOMIAL OF TURÁN 5 Remark.3. An application of Lemmas.1 and. leads to a result of Steinig [1], who proved that the sine polynomial in (.) is nonnegative on (, π), if γ k,n, k = 1,..., n. Lemma.4. Let a k, k = 1,..., n, n 3, be real numbers such that a k a k 1 + a k+1, k =,..., n 1 and a n a n 1. Then, for (, π), (.3) L n () = Proof. Let (, π). We define Then, we have (.4) L n () = Let By assumption, n a j sin(j) a j+ sin(j). c j = a j a j+, j = 1,..., n, c n 1 = a n 1, c n = a n. c j sin(j). γ k = a k a k+1 + a k+, k = 1,..., n, γ n 1 = a n 1 a n, γ n = a n. (.5) γ k, k = 1,..., n. We have n k γ k = c k + ( 1) j c j+k, k = 1,..., n, so that Lemma. implies (.6) c j sin(j) = γ j ϕ j (). Applying Lemma.1 and (.5) reveals that the sum on the right-hand side of (.6) is nonnegative. From (.4) and (.6), we obtain (.3).

6 6 HORST ALZER AND MAN KAM KWONG Remark.5. We assume that L n ( ) = with (, π). The proof of Lemma.4 shows that (i) if n 3 and a n >, then ϕ n ( ) =. Moreover, since ϕ ( ) >, we obtain that (ii) if n = 3, then a a 3 = ; (iii) if n 4, then a a 3 + a 4 =. The net lemma plays an important role in the proofs of Theorems 3.1 and 3. given in the net section. Lemma.6. For all integers n 3 and real numbers a 1, (, π), we have (.7) S n,a () S n,a (). Proof. Let a 1. We define ( ) n + a j ã j =, j = 1,..., n. n j Then, ã j 1 ã j + ã j+1 = j =,..., n 1, n j 1 a(a 1) (n + a j ν), (n j + 1)! ν=1 ã n 1 ã n = a 1, ã n = 1. Since S n,a () S n,a () = n ã j sin(j) ã j+ sin(j), we conclude from Lemma.4 that (.7) holds for (, π). Remark.7. Let n 3, a 1, (, π) and S n,a ( ) = S n,a ( ). Remark.5 implies that a = 1 and ϕ n ( ) =.

7 ON A SINE POLYNOMIAL OF TURÁN 7 3. Trigonometric sums. Our first two theorems show that Turán s inequality (1.1) can be refined if we assume that either n is odd or n is even. Theorem 3.1. For all odd integers n 1 and real numbers a 1, (, π), we have (3.1) S n,a () sin(). Equality holds if and only if n = 1 or n = 3, a = 1, = π/3. Proof. Let n 3 be odd, a 1 and (, π). Applying Lemma.6 gives S n,a () S 1,a () = sin(). Now, we discuss the cases of equality. A short calculation yields that S 3,1 (π/3) = sin(π/3) = 1 3. We assume that S n,a () = sin() = S 1,a (). Case 1. n = 3. Applying Remark.7 leads to a = 1 and ϕ 3 () = sin()(1 + cos()) =. This gives = π/3. Case. n 5. From Lemma.6, we conclude that S 1,a () = S n,a () S 5,a () S 3,a () S 1,a (). Thus, S 3,a () = S 1,a () and S 5,a () = S 3,a (). Applying Remark.7 yields ϕ 3 () = and ϕ 5 () =. The first equation yields = π/3. However, ϕ 5 (π/3) = 3/, a contradiction. It follows that equality holds in (3.1) if and only if n = 1 or n = 3, a = 1, = π/3.

8 8 HORST ALZER AND MAN KAM KWONG Theorem 3.. For all even integers n 1 and real numbers a 1, (, π), we have (3.) S n,a () sin()(1 + cos()). Equality holds if and only if n =, a = 1 or n = 4, a = 1, = π/. Proof. Let a 1 and (, π). Using (.7) gives, for even n, (3.3) S n,a () S,a () = (1 + a) sin() + sin() sin()(1 + cos()). We have and Net, we assume that S,1 () = sin()(1 + cos()) S 4,1 (π/) = sin(π/)(1 + cos(π/)) =. (3.4) S n,a () = sin()(1 + cos()). Case 1. n =. We obtain Thus, a = 1. = S,a () sin()(1 + cos()) = (a 1) sin(). Case. n = 4. Applying (3.3) and (3.4) yields S 4,a () = S,1 (). It follows from Remark.7 that a = 1 and Thus, = π/. ϕ 4 () = 8 sin()(1 + cos()) cos () =. Case 3. n 6. From (.7), (3.3) and (3.4) we conclude that so that Remark.7 gives S 4,a () = S,a () and S 6,a () = S 4,a (), ϕ 4 () = and ϕ 6 () =.

9 ON A SINE POLYNOMIAL OF TURÁN 9 The first equation yields = π/, but ϕ 6 (π/) =. Hence, equality holds in (3.) if and only if n =, a = 1 or n = 4, a = 1, = π/. Remark 3.3. From Theorems 3.1 and 3., we conclude that the estimate (3.5) S n,a () > sin() min{1, (1 + cos())} is valid for n 1, a 1 and (, π). The lower bounds given in (1.3) and (3.5) cannot be compared. Indeed, the function (π ) π sin() min{1, (1 + cos())} is negative on (, 1 ) and positive on ( 1, π), where 1 = The following etension of inequality (1.1) is valid. Theorem 3.4. Let a be a real number. The inequality (3.6) S n,a () > holds for all integers n 1 and real numbers (, π) if and only if a 1. Proof. From (3.5), we conclude that, if a 1, then (3.6) holds for all n 1 and (, π). Conversely, let (3.6) be valid for all n 1 and (, π). Since S,a (π) =, we obtain d d S,a() = 1 a. =π Thus, a 1. Net, we present inequalities for the sine polynomial ( ) n + a j Sn,a() = sin(j). n j j odd Applications of Theorems 3.1 and 3. lead to counterparts of (3.1) and (3.).

10 1 HORST ALZER AND MAN KAM KWONG Theorem 3.5. For all odd integers n 1 and real numbers a 1, (, π), we have (3.7) S n,a() sin(). Equality holds if and only if n = 1. Proof. Let n 1 be odd and a 1, (, π). Inequality (3.1) leads to S n,a() = S n,a () + S n,a (π ) sin() + sin(π ) = sin(). If equality holds in (3.7), then S n,a () = sin() and S n,a (π ) = sin(π ). From Theorem 3.1, we conclude that n = 1. Theorem 3.6. For all even integers n and real numbers a 1, (, π), we have (3.8) S n,a() sin(). Equality holds if and only if n =, a = 1 or n = 4, a = 1, = π/. Proof. Let n be even and a 1, (, π). We apply (3.) and obtain S n,a() = S n,a () + S n,a (π ) sin()(1 + cos()) + sin(π )(1 + cos(π )) = 4 sin(). If n =, a = 1 or n = 4, a = 1, = π/, then equality is valid in (3.8). Conversely, if equality holds in (3.8), then and S n,a () = sin()(1 + cos()) S n,a (π ) = sin(π )(1 + cos(π )). Applying Theorem 3. leads to n =, a = 1 or n = 4, a = 1, = π/.

11 ON A SINE POLYNOMIAL OF TURÁN 11 Now, we study the sine sum in two variables given in (1.4). The following two theorems offer improvements of inequality (1.). Theorem 3.7. For all odd integers n 1 and real numbers a 1,, y (, π), we have (3.9) Θ n,a (, y) sin() sin(y). Equality holds if and only if n = 1. Proof. Let n 1 be odd and a 1. Since equality holds in (3.9), if n = 1, we suppose that n 3. Applying Theorem 3.1 gives for, y R with < y < π and < + y < π: S n,a ( y) + S n,a ( + y) sin( y) + sin( + y). This leads to (3.1) ( ) n + a j sin(j) cos(jy) sin() cos(y). n j Equality holds in (3.1) if and only if n = 3, a = 1 and y = π/3, + y = π/3, that is, = π/3, y =. In order to obtain (3.9) we integrate both sides of (3.1) with respect to y. Let, y R with < y < π. We consider two cases. Case 1. π/. Let < y < y. Then, < y < + y < π. It follows from (3.1) with > instead of that (3.11) y ( ) n + a j y sin(j ) cos(jy) dy > sin( ) cos(y) dy. n j This leads to (3.9) with =, y = y and > instead of. Case. π/ <. Case.1. y π. Let < y < y. Then, < y < + y < π, so that we obtain (3.11) and (3.9) with =, y = y and > instead of. Case.. π < y. We set 1 = π and y 1 = π y. Let < y < 1. Then, < y 1 y < y 1 + y < π. This leads to (3.11) with

12 1 HORST ALZER AND MAN KAM KWONG 1 instead of y and y 1 instead of. Using sin(jy 1 ) = ( 1) j 1 sin(jy ), j N, and sin(j 1 ) = ( 1) j 1 sin(j ), j N, we obtain (3.9) with =, y = y and > instead of. Theorem 3.8. For all even integers n and real numbers a 1,, y (, π), we have (3.1) Θ n,a (, y) sin() sin(y)(1 + cos() cos(y)). Equality holds if and only if n =, a = 1. Proof. The proof is similar to that of Theorem 3.7. Therefore, we only offer a proof sketch. Since ( ) a + 1 Θ,a (, y) = sin() sin(y) + cos() cos(y), we conclude that, if n =, then (3.1) is valid with equality if and only if a = 1. Let n 4. Using Theorem 3., we obtain for, y R with < y < π and < + y < π: (3.13) ( ) n + a j sin(j) cos(jy) sin() cos(y) + sin() cos(y). n j Equality holds in (3.13) if and only if n = 4, a = 1, = π/, y =. Let, y R with < y < π. We assume that π/. If < y < y, then < y < y < π. Net, we integrate both sides of (3.13) (with = and > instead of ) from y = to y = y. This gives (3.14) Θ n,a (, y ) > sin( ) sin(y )(1 + cos( ) cos(y )). By similar arguments, we conclude that (3.14) also holds if > π/.

13 ON A SINE POLYNOMIAL OF TURÁN 13 Let Θ n,a(, y) = ( ) n + a j sin(j) sin(jy). n j j j odd Applying Theorems 3.7 and 3.8 we obtain the following companions of (3.7) and (3.8). Theorem 3.9. For all real numbers a 1 and, y (, π), we have (3.15) and (3.16) Θ n,a(, y) sin() sin(y), Θ n,a(, y) sin() sin(y), if n is odd if n is even. Equality holds in (3.15) if and only if n = 1 and in (3.16) if and only if n =, a = 1. The proof of this theorem is quite similar to the proofs of Theorems 3.5 and 3.6; therefore, we omit the details. We conclude this section with a generalization of inequality (1.). Theorem 3.1. Let a be a real number. The inequality (3.17) Θ n,a (, y) > holds for all integers n 1 and real numbers, y (, π) if and only if a 1. Proof. Let a 1. From Theorems 3.7 and 3.8 we conclude that Θ n,a (, y) is positive for all n 1 and, y (, π). Conversely, if (3.17) holds for all n 1 and, y (, π), then we have Θ,a (, y) = and Θ,a(, y) = sin(y)(1 + a + cos(y)). = It follows that 1 + a + cos(y). We let y π and obtain a 1.

14 14 HORST ALZER AND MAN KAM KWONG 4. Chebyshev polynomials. The Chebyshev polynomials of the first and second kind, T n () and U n (), n =, 1,,..., are polynomials in of degree n, defined by and (4.1) U n () = T n () = cos(nt) sin((n + 1)t), = cos(t), t [, π]. sin(t) They have remarkable applications in numerical analysis, approimation theory and other branches of mathematics. The main properties of these functions may be found, for instance, in Mason and Handscomb [8]. Using the notation (4.1), we obtain from (1.1) the inequality ( ) n + k j (4.) Λ n,k () = U j () >, n j j= n Z, k N, 1 < < 1. The results presented in Section 3 lead to inequalities for Chebyshev polynomials of the second kind. As eamples, we offer counterparts of Theorems 3.1 and 3. which provide refinements of (4.). Theorem 4.1. For all even integers n and real numbers a 1, ( 1, 1), we have Λ n,a () 1. Equality holds if and only if n = or n =, a = 1, = 1/. Theorem 4.. For all odd integers n 1 and real numbers a 1, ( 1, 1), we have Λ n,a () (1 + ). Equality holds if and only if n = 1, a = 1 or n = 3, a = 1, =. The inequalities (4.3) < sin(j) j < π, n N, < < π,

15 ON A SINE POLYNOMIAL OF TURÁN 15 are classical results in the theory of trigonometric polynomials. In 191, Fejér conjectured the validity of the left-hand side. The first proofs were published by Jackson [7] in 1911 and Gronwall [6] in 191. The righthand side is due to Turán [13], who proved his inequality in The following interesting companion of (4.3) was given by Carslaw [4] in 1917: (4.4) < j= sin((j + 1)) j + 1 1, n Z, < < π. Equality holds if and only if n =, = π/ (also see []). We show that applications of (4.3) and (4.4) lead to integral inequalities for the Chebyshev polynomials. Theorem 4.3. For all integers n 1 and real numbers (, 1), we have (4.5) arccos() < U n (t) dt < π arccos(). If ( 1, ), then (4.5) holds with > instead of <. Proof. Let t (, π) and Then, F n(t) = F n (t) = cos(jt) = This gives, for z (, π): F n (z) = z F n(t) dt = z sin(jt). j sin((n + 1)t) sin(t) U n (cos(t)) dt z = 1 = U n (cos(t)) 1. cos(z) If (, 1), then arccos() (, π/). Using (4.3) leads to < F n (arccos()) = This implies (4.5). U n (t) dt z. U n (t) dt arccos() < π arccos().

16 16 HORST ALZER AND MAN KAM KWONG We define for ( 1, 1): G n () = arccos() Applying (4.6) w() + w( ) = U n (t) dt and w() = arccos() π. and yields G n ()+G n ( )= ( U n (t) dt = π Since U n is an even function, we obtain Thus, U n (t) dt = ) ( Un (t) dt = 1 t U n ( t) dt = (4.7) G n () + G n ( ) =. Using (4.6) and (4.7) gives that the function satisfies H n () = G n () w() (4.8) H n () + H n ( ) =. U n (t) dt. ) Un (t) 1 t dt. From (4.5), (4.7) and (4.8), we conclude that, for ( 1, ), we have G n () = G n ( ) < < H n ( ) = H n (). This leads to (4.5) with > instead of <. Remark 4.4. The functions U n (t) dt arccos(), n = 3, 7, 11,

17 ON A SINE POLYNOMIAL OF TURÁN 17 and U n (t) π arccos() dt, n = 1, 5, 9, attain positive and negative values on (, 1). This implies that in general Theorem 4.3 is not true for Chebyshev polynomials of odd degree. We conclude this paper with a theorem which offers sharp upper and lower bounds for an integral involving the Chebyshev polynomials of the first and second kind. Theorem 4.5. For all integers n and real numbers ( 1, 1), we have (4.9) < T n+1 (t) U n (t) dt 1. Equality holds on the right-hand side if and only if n =, =. Proof. Let ( 1, 1). Since T 1 (t) = t and U (t) = 1, we obtain T 1 (t)u (t) This leads to (4.9) with n =. dt = 1. Let n 1. We denote the sine sum in (4.4) by I n (). Then, for t (, π), I n(t) = cos((j + 1)t) = j= = T n+1 (cos(t)) U n (cos(t)). Hence, we obtain for z (, π): I n (z) = z I n(t) dt = cos((n + 1)t) sin((n + 1)t) sin(t) cos(z) so that (4.4) (with < instead of ) gives < I n (arccos()) = T n+1 (t) U n (t) T n+1 (t) U n (t) dt < 1. dt,

18 18 HORST ALZER AND MAN KAM KWONG Acknowledgments. We thank the referee for a careful reading of the paper and for helpful comments. REFERENCES 1. H. Alzer and B. Fuglede, On a trigonometric inequality of Turán, J. Appro. Th. 164 (1), H. Alzer and S. Koumandos, On the partial sums of a Fourier series, Constr. Appro. 7 (8), H. Alzer and M.K. Kwong, Etension of a trigonometric inequality of Turán, Acta Sci. Math. 8 (14), H.S. Carslaw, A trigonometrical sum and Gibbs phenomenon in Fourier s series, Amer. J. Math. 39 (1917), L. Fejér, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen;..., Monatsh. Math. Phys. 35 (198), T.H. Gronwall, Über die Gibbssche Erscheinung und die trigonometrischen Summen sin + 1 sin sin n, Math. Ann. 7 (191), n 7. D. Jackson, Über eine trigonometrische Summe, Rend. Circ. Mat. Palermo 3 (1911), J.C. Mason and D.C. Handscomb, Chebyshev polynomials, Chapman and Hall, Boca Raton,. 9. G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in polynomials: Etremal problems, inequalities, zeros, World Scientific, Singapore, J. Steinig, A criterion for the positivity of sine polynomials, Proc. Amer. Math. Soc. 38 (1973), G. Szegő, Power series with multiply sequences of coefficients, Duke Math. J. 8 (1941), P. Turán, Über die arithmetischen Mittel der Fourierreihe, J. Lond. Math. Soc. 1 (1935), , Über die Partialsummen der Fourierreihe, J. Lond. Math. Soc. 13 (1938), Morsbacher Str. 1, Waldbröl, Germany address: h.alzer@gm.de The Hong Kong Polytechnic University, Department of Applied Mathematics, Hunghom, Hong Kong address: mankwong@connect.polyu.edu.hk