Browsing by Author "Alzer, H."

Now showing items 1-18 of 18

    • Article  

      Companions of the inequalities of Fejér-Jackson and young 

      Alzer, H.; Koumandos, S. (2005)
      Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities, We prove that the following counterpart is valid for all integers n ≥ 1 and real numbers x ∈ (0, π): where the sign of equality ...
    • Article  

      Inequalities for two sine polynomials 

      Alzer, H.; Koumandos, S. (2006)
      We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0, π) we have (Formula Presented) with the best possible constant bounds (Formula Presented) (II) The inequality (Formula Presented) holds for all even integers n ...
    • Article  

      Inequalities of Fejér-Jackson Type 

      Alzer, H.; Koumandos, S. (2003)
      We complement, extend, and sharpen some known inequalities for sine sums. Our main result is the following refinement of the classical Fejér-Jackson inequality: For all integers n ≥ 2 and real numbers x ∈ (0, π) we have ...
    • Article  

      A new refinement of Young's inequality 

      Alzer, H.; Koumandos, S. (2007)
      A classical theorem due to Young states that the cosine polynomial c n(x) = 1 + ∑k=1n cos(kx)/k is positive for all n ≥ 1 and x ∈ (0, π). We prove the following refinement. For all n ≥ 2 and x ∈ [0, π] we have 1/6 + c(π - ...
    • Article  

      On a conjecture of Clark and Ismail 

      Alzer, H.; Berg, C.; Koumandos, S. (2005)
      Let Φm (x) = -xmψ(m) (x), where ψ denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m ∈ {1,2,..., 16}, then Φm(m) is completely monotonic on (0, ...
    • Article  

      On a trigonometric sum of Vinogradov 

      Alzer, H.; Koumandos, S. (2004)
      The trigonometric sum f(m, n) = ∑ k=1 m-1 sin(πkn/m) /sin(πk/m) (1 < m ∈ N, n ∈ N) has several applications in number theory. We prove that the mean value inequalities c1m(log m + γ - logπ/2) ≤ 1/m ∑ n=1 f(m, n) < c2m(log ...
    • Article  

      On the partial sums of a fourier series 

      Alzer, H.; Koumandos, S. (2008)
      We give sharp lower estimates for the partial sums of the Fourier series sin x + cos2x/2 + sin 3x/3 + cos 4x/4 + ..., with both an even and odd number of terms. Our results are obtained through a monotonicity property of ...
    • Article  

      A refinement of Vietoris' inequality for sine polynomials 

      Alzer, H.; Koumandos, S.; Lamprecht, M. (2010)
    • Article  

      Remarks on a sine polynomial 

      Alzer, H.; Koumandos, S. (2009)
      Let n ≥ 0 be an integer. Then we have for x ε (0, π) : ∑k=0 n(2n+1 n-k)sin((2k+1)x)/2k+1 ≤ 8 n-rfnet-temp!/(2n+1)!! The upper bound is best possible. This complements a result of Fejér, who proved that the sine polynomial ...
    • Article  

      Series and product representations for some mathematical constants 

      Alzer, H.; Koumandos, S. (2009)
      We present several series and product representations for γ, π, and other mathematical constants. One of our results states that, for all real numbers μ s>0, we have γ = ∑k=0∞ 1/(1 + μ)k+1∑m=0k,(m k)} (-1) mμk-m S(m), where ...
    • Article  

      Series representations for γ and other mathematical constants 

      Alzer, H.; Koumandos, S. (2008)
      We present series representations for some mathematical constants, like γ, π, log 2, ζ(3). In particular, we prove that the following representation for Euler's constant is valid: Equation presented. © 2008 Springer ...
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      A sharp bound for a sine polynomial 

      Alzer, H.; Koumandos, S. (2003)
      We prove that (Formula Presented) for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951). © 2003, Instytut Matematyczny. All ...
    • Article  

      Sharp estimates for various trigonometric sums 

      Alzer, H.; Koumandos, S. (2012)
      We offer sharp inequalities for trigonometric sums of the classical Fejér-Jackson-Young-type, and we present sharp bounds for alternating trigonometric sums involving binomial coefficients. Among others, we prove. © 2012, ...
    • Article  

      Sharp inequalities for trigonometric sums 

      Alzer, H.; Koumandos, S. (2003)
      We prove the following two theorems: (I) Let n ≥ 1 be a (fixed) integer. Then we have for θ ∈ (0, π): ∑k=1ncos(kθ)/k ≤ -log (sin (θ/2)) + π-θ/2 + σn, with the best possible constant σn = ∑k=1n (-1)k/k. (II) For even integers ...
    • Article  

      Sharp inequalities for trigonometric sums in two variables 

      Alzer, H.; Koumandos, S. (2004)
      We prove several new inequalities for trigonometric sums in two variables. One of our results states that the double-inequality -2/3(√2 - 1) ≤ Σk=1n cos((k - 1/2)x) Sin((k - 1/2)y)/k - 1/2 ≤ 2 holds for all integers n ≥ 1 ...
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      A Sharp Inequality for a Trigonometric Sum 

      Alzer, H.; Koumandos, S. (2013)
      We prove that the inequality, holds for all natural numbers n and real numbers x with x ∈ [0, Π]. The sign of equality is valid if and only if n = 1 and x = π /2. © 2012 Springer Basel AG.
    • Article  

      Some monotonic trigonometric sums 

      Alzer, H.; Koumandos, S. (2007)
      Let be the trigonometric polynomials of Fejer and Young, respectively. The aim of this paper is to determine all real parameters a and a such that for all integers n > 1 the functions. © 2007, Oldenbourg Wissenschaftsverlag. ...
    • Article  

      Sub- and superadditive properties of Fejér's sine polynomial 

      Alzer, H.; Koumandos, S. (2006)
      Let Sn(x) = ∑k=1n (sin(kx))/k be Fejér's sine polynomial. We prove the following statements. (i) The inequality (Sn (x + y))α (x + y)β ≤ (Sn (x))α xβ + (Sn (y))αybeta