#### 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
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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.
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