ASYMPTOTIC BEHAVIOR FOR THE BEST LOWER BOUND OF JENSEN S FUNCTIONAL

Transcription

1 75 Kragujevac J. Math ) ASYMPTOTIC BEHAVIOR FOR THE BEST LOWER BOUND OF JENSEN S FUNCTIONAL Stojan Radenović and Mirjana Pavlović 2 University of Belgrade, Faculty of Mechanical Engineering, 27. marta 8, Belgrade, Serbia and Montenegro 2 University of Kragujevac, Faculty of Science, Deartment of Mathematics, P.O.Box 6, 34 Kragujevac, Serbia and Montenegro Received August 5, 22) Abstract. In this note we consider asymtotic estimates for the best lower bound of Jensen s functional f log fe iθ ) dθ, f satisfies ) and 2) below, when k +. Let fz) = n a j z j ) be a olynomial with comlex coefficients and let d be j= a real number such that < d <. Then fz) is said to have concentration d at degrees at most k, measured by the l -norm, 2), if a j d a j. ) j Polynomials with concentrations of low degrees introduced by B.Beauzamy and P.Enflo, who roved, for such olynomials, a generalized Jensen s inequality [] and [2]; this lays an imortant role in the construction of an oerator on a Banach sace with no non-trivial invariant subsace [4].

2 76 We investigate here the estimates of Jensen s functional f log ) fe iθ ) dθ f l for olynomials satisfying ).In the sequel, we shall normalize f and assume that a j j =. 2) For such olynomials, it is shown in [6] the following: If fz) = n a j z j is a olynomial which satisfies ) and 2), then: j= log fe iθ ) dθ Cd, k, ) = max f d,k,t), 3) <t<+ where t f d,k, t) = log d [ t+ t ) ] t+ t )k+) 2 t2, < 2 2d t log, =. t )[ t+ t )k+ ] For our urose here, Cd, k, ) will denote the largest such constant ossible in 3), i.e. Cd, k, ) := inf log fe iθ ) dθ : f satisfies ) and 2). 4) In the sequel, we have restricted ourselves to the numerical asymtotic estimates for Cd, k, ), when k +, < d < and < 2. Theorem. Let ], 2] and d ], 2 ]. Then for sufficiently large k the best constant Cd, k, ) satisfies : 2k Cd, k, ) 2k log 2. where Proof. Firstly, we reresent f d,k, t) in the form: f d,k, t) = h d, t) + g k t) t log [ ) t k+) ], t + h d, t) = t log d 2 t2 + t log[t + ) t ) ] g k t) = kt logt ) k + )t logt + ). It is clear that f d,k, t) > h d, t) + g k t), t >. We shall now rove that the function h d, t) + g k t) takes its maximum value at a oint unique) t k such that

3 77 t k +, when k +. We shall maximize h d, t) + g k t), since the remaining [ term t log ) ] k+) t can be neglected, for t = t t+ k, when k +. We now find derivatives for h d, t) and g k t). g kt) = k + ) logt + ) + k logt ) g kt) = t )2 t + 2) + 4k t 2 ) 2 < ; k + )t t + + kt t ; h d,t) = log d t + log[t + ) t ) ] + t t + ) t ) t + ) t ) ; h d,t) = + 2 t + ) t ) t + ) t ) t ) [t + + ) 2 t ) 2] At) [ t + ) t ) ] 2 ), A 2 t) where At) = t + ) t ). Since ], 2], t >, it is clear that h d, < iff + 2 t + ) t ) At) But, this is true iff ϕ t) <, where ϕ t) = 2t+) 2t ) t+) +t ). Hence, we find that <. ϕ t) = 2 ) [ t + ) 2 t ) 2)] + [ t ) t + ) ] <. This shows that h d,t) + g kt) <. Since lim t + h d,t) + g kt)) = + and lim t + h d,t) + g kt)) =, equation h d,t) + g kt) = has exactly one solution t k. From the equality h d,t) + g kt) = we get with t = t k, k = t2 ) logt + ) + t 2 t t 2 )h d,t) 2t + t 2 ) logt ) t 2 ) logt + ), 5) wherefrom we easily deduce that t k + iff k +. Writing logt ± ) = log t + log ± t ), t ± ) = t ± t ) and substituting Taylor exansion of order 3 when k +, we get k 3 4 t4 k. 6)

4 78 So we have shown that at the oint t k, the value of f d,k, t) and the value of h d, t) + g k t) are asymtotically the same. All we have to do now is to comute f d,k, t k ), and this follows easily by the estimate 6) and the asymtotic equalities for log ± t ) and ± t ) : f d,k, t k ) h d, t k ) + g k t k ) = ) 2 t2 k + o)) + 2k) + o)) ) = 2k) + o) + t2 k 4k = 2k) + o)) 2k, because t2 k 4k t2 k =, k +. This roves the asymtotic estimate t4 3t 2 k k Cd, k, ) f d,k, t k ) 2k, k +, for each d ], [ and for each ], 2]. If =, from [2] it follows that k 3 4 t3 k log t k, k +. Consider now the olynomial gz) = g g z) where g z) = + z l 2 2 )2k+. It satisfies ) and 2). By the roerties of the binomial coefficients, this olynomial has concentration d 2 at degrees k, measured by the l -norm. Indeed, from 2k + g l 2 2k+) j ) d j ) 2k + g, l 2 2k+) j it follows ) 2k + d 2 ) 2k +, 2 2k+) j 2 2k+) j that is < d 2. But the constant term is g l 22k+), the only root is, so Jensen s formula says that: log ge iθ ) dθ = 2k + ) log 2 log g l = 2k log 2) + o)) 2k log 2. Hence, we have asymtotically: Cd, k, ) 2k log 2, when k +, for each d ], 2 ] and for each ], 2]. The same estimate is true and in case = see[2]). Remark. If k =, it follows that f d,, t) = t log d 2 t2, < 2, as and f d,, t) = t log d. Now, we obtain that Cd,, ) = log d, < 2, i.e. 2

5 79 Cd,, ) = log d. Hence, there exists the recise value of the best constant Cd,, ). This generalizes Theorem from [2]. Our Theorem also generalizes the corresonding result in [], Lemme 3) for = 2. It should be noted that in case k >, nothing is known about the recise value of Cd, k, ), even for small values of k. Acknowledgements: This research was suorted by Ministry of Science, Technology and Develoment of Serbia, roject Structures of Functional Analysis and Differential Equations, no References [] B.Beauzamy et P.Enflo, Estimations de roduits de olynômes, J.Number Theory 2985), [2] B.Beauzamy, Jensen s inequality for olynomials with concentration at low degrees, Numer. Math ), [3] B.Beauzamy, A Minimization Problem Connected with a Generalized Jensen s Inequality, Journal of mathematical analysis and alications 4599), [4] P.Enflo, On the invariant subsaces roblem in Banach saces, Acta Math ), [5] M.Pavlović, Asimtotsko onašanje donjih granica Jensenovog funkcionala, Magistarska teza, PMF, Kragujevac, 998). [6] S.Radenović, Some estimates of the integral Beograd) 5266)992), log ge iθ ) dθ, Publ. Inst. Math. [7] S.Radenović,A lower bound of Jensen s functional, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 5994), 9-2.