Improved estimates for the triangle inequality

Journal of Inequalities and Applications, Jan 2017

We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail. MSC: 46B99, 26D15, 46C50, 46C05.

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Improved estimates for the triangle inequality

Minculete and Pa˘lta˘nea Journal of Inequalities and Applications Improved estimates for the triangle Nicus¸or Minculete 0 1 2 Radu Pa˘lta˘nea 0 1 2 0 University of Bras ̧ov , Str. Iuliu Maniu 1 Computer Science , Transilvania 2 Faculty of Mathematics We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail. MSC: Primary 46B99; secondary 26D15; 46C50; 46C05 b for any vectors x and y in the normed linear space (X, · ) over the real numbers or complex numbers. Its continuous version is, where f : [a, b] ⊂ R → X is a strongly measurable function on the compact interval [a, b] with values in a Banach space X and f (·) is the Lebesgue integrable on [a, b]. norm inequalities; triangle inequality; Tapia semi-product; inner product 1 Introduction The theory of inequalities plays an important role in many areas of Mathematics. Among the most used inequalities we find the triangle inequality. This inequality is the following: x + y ≤ x + y , f (t) dt ≤ ≤ x + y – x + y © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In [] Kato, Saito and Tamura proved the sharp triangle inequality and reverse inequality in Banach space for nonzero elements x, x, . . . , xn ∈ X, which is in fact a generalization of Maligranda’s inequality. Another extension of Maligranda’s inequality for n elements in a Banach space was obtained in Mitani and Saito []. The problem of characterization of all intermediate values C satisfying  ≤ C ≤ kn= xk – kn= xk , for x, x, . . . , xn in a Banach space is studied by Mineno, Nakamura and Ohwada [], Dadipour et al. [], Sano et al. [] and others. For other different results about the triangle inequality we mention only [–]. The main aim of the present paper is to provide an improvement of the inequality due to Maligranda. Some other estimates which follow from the triangle inequality are also presented. Moreover, we can rewrite them as estimates for the so-called norm-angular distance or Clarkson distance (see e.g. []) between nonzero x and y as α[x, y] = xx – yy . This distance was generalized to the p-angular distance in normed space in []. In [], Dragomir characterizes this distance obtaining new bounds for it. A survey on the results for bounds for the angular distance, so named Dunkl-Williams type theorems (see []), is given by Moslehian et al. []. 2 Estimates of the triangle inequality using integrals Let (X, · ) be a real normed space. The following lemma is evident. Theorem  For any x, y ∈ X we have (i) x + y ≤   ( – λ)x + λy dλ ≤ x + y , (ii) x + y + x + y ≥   ( – λ)x + λy dλ. Therefore we obtain z + su ds ≤ For any a < b and any x, y ∈ X, u = a–b (x – y) and z = a–b (ay – bx) such that x = z + au and y = z + bu. By replacing in inequality () we deduce the inequality bx – ay + s(y – x) ds ≤ If we multiply these inequalities by  and make the change of variable s = ( – λ)a + λb in the integral, we arrive at relation (i). (ii) If we apply the Hammer-Bullen inequality, see e.g. [], for a < b we obtain Now, if we proceed similarly to (i) we obtain relation (ii). Example  Let X = R, endowed with the norm (x, x) = max{|x|, |x|}. Let η, μ ∈ [, ] and define x = (, ), y = (–μ, ημ). We have x = , y = μ, x + y =  + ημ. For λ ∈ [, ], since ( – λ)(–λη) ≤  and ( – λ)λημ ≥ , we obtain ( – λ)x + λy = ( – λ – λμ,  – λ + λημ) =  – λ + λημ and then Hence, relations (i) and (ii) become Remark  Since the parameters η, μ can be taken arbitrarily in the interval [, ] in Example , it follows that the constants in front of the terms of inequalities (i) and (ii) in Theorem  are optimal. Corollary  For nonzero elements x, y from a space with inner product X = (X, ·, · ) and any a, b ∈ R, a < b, we have ( x y – x, y ) x + y +   ( – λ)x + λy dλ ≤ x + y – x + y x + y  = which means that Using point (i) from Theorem  in the above equality we obtain the inequalities of the statement. Remark  Inequality () represents an improvement of the Cauchy-Schwarz inequality. 3 Estimates of the triangle inequality using the Tapia semi-product The Tapia semi-product on the normed space X (see []) is the function (·, ·)T : X × X → R, defined by (x, y)T = ltim ϕ(x + tyt) – ϕ(x) , where ϕ(x) =  x , x ∈ X. The above limit exists for any pair of elements x, y ∈ X. The Tapia semi-product is positive homogeneous in each argument and satisfies the inequality if x = , where (x, y)T = For nonzero elements x, y ∈ X denote First from the Maligranda result we deduce the following inequality. (x, y)T ≤ x · y v(x, y) –  . Proof If in the left inequality of Theorem A we replace y by ty, with t > , t < x / y and then we divide them by t we obtain  – v(x, y) y ≤ y – Letting t →  we obtain  – v(x, y) From this we obtain immediately equation (). Remark  Inequality () is an improvement of the known inequality (x, y)T ≤ x · y , since v(x, y) ≤ . x + y – x + y ≥  – v(x, y) x + y – x + y ≤  – v(x, y) x + y – x + y ≥  – v(x, y) x + y – x + y ≤  – v(x, y) Proof We choose the following notations: f (s) =  + s – z + su , s ∈ [, ∞). g(s) = z + su , s ∈ [, ∞). = g (s + ), where we denoted by g (s + ) the right derivative of g at point s. Then f (s + ) =  – , s ∈ [, ∞). f () – ( – λ)f (λ + ) ≤ f (λ) ≤ f () – ( – λ)f ( + ). Because these inequalities are obvious for λ = , we suppose that  ≤ λ < . Choose an arbitrary number t > . The function f is the difference between a linear and a convex function. Then it is concave. Consequently we have tt –– λ f (λ) + t –– λλ f (t) ≤ f (). Straightforward computations shows that this inequality can be written equivalently, in the form f (λ) ≤ f () – ( – λ) f (tt) –– f() . t –– λλ f () +  –– λt f (λ) ≤ f (t). x f (λ) = x + y – x + y , x f () = x  – v(x, y) , f(s) =  + s – sz + u , s ∈ [, ∞). If we pass to the limit t → , t >  in this inequality we obtain the right side inequality in (). Also, choose an arbitrary number λ < t < . Since f is concave we have After short computations, this inequality can be written equivalently, in the form Passing to the limit t → λ, t > λ we obtain the left side inequality in (). From equation () we obtain f ( + ) =  – If we multiply equations () by x and we take into account relations () and also the following relations: we deduce inequalities () and (). In order to obtain the last two inequalities we consider the function f : [, ∞) → R, The right derivatives of the function f can be obtained immediately from the derivatives of the function f by interchanging u and z, see equation (). So we obtain f (s + ) =  – , s ∈ [, ∞). We consider only the case μ > , since for μ =  these inequalities are obvious. First take an arbitrary number t > μ. Since f is concave, we have This inequality can be rewritten in the equivalent form f() + f(tt) –– fμ(μ) (μ – ) ≤ f(μ). If we pass to the limit t → μ, t > μ we obtain the left side inequality in (). Next let us take an arbitrary number  < t < μ. Since f is concave we have μt–– f(μ) + μμ –– t f() ≤ f(t). We can transform this inequality to the form f(μ) ≤ f() + f(tt) –– f() (μ – ). If we pass to the limit t → , t >  we obtain the right side inequality in (). If we multiply inequalities given in () by y and take into account the relations y f(μ) = x + y – x + y , y f() = y  – v(x, y) , f ( + ) =  – we obtain relations () and (). Remark  Inequalities () and () improve inequality (). Example  Let the space X and the vectors x, y be exactly like in Example . Then we have x + y – x + y = μ( – η). Also v(x, y) = (, ) + (–, η) = (,  + η). Then ( – v(x, y) ) x =  – η. Hence the right inequality in () of Theorem A reads μ( – η) ≤  – η. We obtain an improvement of this inequality by using inequality () given in Theorem . Indeed we have = (, ), (–, η) T = ltim t (–t,  + tη)  – (, )  = η. It is easy to see that we can write α[x, y] = v(x, –y) . Using inequalities () and () we deduce the following double inequality. x –myin{– x| x, y– } y | + A ≤ α[x, y] ≤ A =  – B =  – m|inx{ x– , yy| } ≥ , m|axx{ x– , y y| } ≥ . Remark  Inequalities () improve the inequalities for the norm-angular distance, of Maligranda [], which can be obtained from (). Other inequalities for the norm-angular distance could be obtained combining all inequalities (), (), (), and (). 4 Inequalities in inner product spaces In this section we derive inequalities in an inner product space (X, ·, · ) from Theorem , by taking into account that (x, y)T = x, y , x, y ∈ X. Theorem  Let (X, ·, · ) be an inner product space, with norm · . For nonzero elements x, y ∈ X v(x, y) = v(x, y)  = =   + Hence we get (). Next we obtain We can apply Theorem . From equation (), by taking into account equation () we obtain x + y – x + y ≤  – v(x, y) =  – v(x, y) Remark  From the proof we can see that, in an inner product space, inequality () is equivalent to inequality (). In a similar way we can obtain  – v(x, y) =  – v(x, y) This means that, if X is an inner product space, inequality () is also equivalent to inequality (). Thus in an inner product space equations () and () are equivalent. Remark  Inequality () can be written equivalently on the form By changing y by –y in () and taking into account that α[x, y] = v(x, –y) , we see that we have the Dunkl-Williams inequality in an inner product space; see [, ]. Inverse inequalities are given in the next theorem. Theorem  Let (X, ·, · ) be an inner product space, with norm ments x, y ∈ X be such that x + y = . (i) If x ≥ y , then (ii) Without condition x ≥ y , x + y – x + y ≥ · and let nonzero eleProof (i) We can apply Theorem . From equation () we obtain So we proved inequality (). We apply Theorem . From equation () we deduce that So we proved inequality () too. (ii) By virtue of the symmetry of equation () we can suppose that x ≥ y . If we add inequalities () and () and then we divide by  we obtain Remark  From the proof of Theorem  it follows that in an inner product space equation () is equivalent to equation () and equation () is equivalent to equation (). Remark  Inequality () has the advantage of the symmetry, but in general it does not improve the triangle inequality. Remark  We note also that inequality () is equivalent to inequality (). Indeed, equation () can be written in equivalent form, x · x + y  ≤ +  ≤ v(x, y) · x + y , inequality () is reduced to an equality. Also inequality () is equivalent to inequality (). Indeed, equation () can be written in equivalent form, ≤ v(x, y) · From this, we reason similarly to in the case of equation (). In conclusion, all relations (), (), (), (), and consequently also (), are equivalent an inner product space inequalities (), (), (), and () are equivalent to each other, for x ≥ y >  and x y = – y x. Competing interests The authors declare that they have no competing interests. Authors’ contributions The work was carried out in collaboration between the two authors. The authors had equal contributions in writing this article. The author NM plays the role of corresponding author. All authors read and approved the final manuscript. Acknowledgements The authors were grateful to the referee for useful comments. 1. Diaz , JB, Metcalf, FT: A complementary triangle inequality in Hilbert and Banach spaces . Proc. Am. Math. Soc. 17 , 88 - 97 ( 1966 ) 2. Dragomir , SS: Reverses of the triangle inequality in Banach space . J. Inequal. Pure Appl. Math . 6 ( 5 ), Article ID 139 ( 2005 ) 3. 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Nicuşor Minculete, Radu Păltănea. Improved estimates for the triangle inequality, Journal of Inequalities and Applications, 2017, 17, DOI: 10.1186/s13660-016-1281-z