Inequality for power series with nonnegative coefficients and applications

(1)

Open Mathematics

Open Access

Research Article

Silvestru Sever Dragomir*

Inequality for power series with nonnegative

coefficients and applications

DOI 10.1515/math-2015-0061

Received July 20, 2015; accepted September 10, 2015.

Abstract:We establish in this paper some Jensen’s type inequalities for functions defined by power series with nonnegative coefficients. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

Keywords: Jensen’s inequality, Measurable functions, Lebesgue integral, Selfadjoint operators, Functions of selfadjoint operators

MSC:26D15, 26D20, 47A63

1 Introduction

In 1906, J. L. W. V. Jensen [13] has proved the following remarkable inequality

f Pn

iD1pixi Pn

iD1pi

Pn

iD1pif .xi/ Pn

iD1pi

(1)

for a convex functionf WI R!Rin the intervalI, elementsxi 2Iand nonnegative numberspi; i_{2 f}1; :::; ng

withPn

iD1pi > 0:This inequality is important in various fields of mathematics due to the fact that one can obtain

from it other important inequalities such as the triangle inequality, Hölder’s inequality, Ky Fan’s inequality, to name only a few.

Let.;A; /be a measurable space consisting of a set;a – algebraAof subsets ofand a countably additive and positive measureonAwith values inR[ f1g:Assume, for simplicity, thatRdD1:Consider

theLebesgue space

L .; /WD ff W!R; f is-measurable and

Z

jf .t /jd .t / <1g:

For simplicity of notation we write everywhere in the sequelRwdinstead ofRw .t / d .t / :

In order to provide a reverse of the celebrated Jensen’s integral inequality for convex functions, the author obtained in [4] and [7] the following result:

Theorem 1.1. LetˆWŒm; M R!Rbe a differentiable convex function on.m; M /andf W!Œm; M so

thatˆıf; f; ˆ0ıf; .ˆ0ıf / f 2L .; / :Then we have the inequality:

0

Z

ˆıf d ˆ

0

@ Z

f d

1

A Z

ˆ0ıf f d

Z

ˆ0ıf d Z

f d (2)

(2)

1 2

ˆ0.M / ˆ0.m/ Z

ˇ ˇ ˇ ˇ ˇ ˇ f

Z

f d

ˇ ˇ ˇ ˇ ˇ ˇ

d

1

2

ˆ0.M / ˆ0.m/ 2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5

1 2

1

4

ˆ0.M / ˆ0.m/

.M m/ :

Remark 1.2. We notice that the inequality between the first and the second term in (2) in the discrete case was proved in 1994 by Dragomir & Ionescu, see [9].

For other recent reverses of Jensen inequality and applications to divergence measures see [6], [7] and [8].

Motivated by the above results we establish in this paper some Jensen’s type inequalities for functions defined by power series with nonnegative coefficients. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

2 Results

The most important power series with nonnegative coefficients are:

exp.z/D 1 X

nD0 1 nŠz

n_{; z} 2C,

1

1 z D

1 X

nD0

zn; z2D .0; 1/ ; (3)

ln 1

1 z D

1 X

nD1 1 nz

n_{; z}

2D .0; 1/ ; coshzD 1 X

nD0 1 .2n/Šz

2n_{; z} 2C,

sinhzD 1 X

nD0 1 .2nC1/Šz

2nC1_{; z} 2C,

1 2ln

₁ Cz

1 z

D

1 X

nD1 1 2n 1z

2n 1_{; z}

2D .0; 1/ ;

where byD .0; R/we denote the open disk centered in0with radiusR > 0:

The following results that improve Jensen inequality as well as provide some reverse inequalities can be stated:

Theorem 2.1. Letˆ .z/DP1

nD0anznbe a power series with nonnegative coefficients and convergent onD .0; R/

withR > 0orRD 1:Assume thatf W!Ris-measurable and with0 < f .u/ < Rfor-almost everyuin

and such thatˆıf; .ˆ0ıf / f; .ˆ0ıf / f 12L .; / :Then we have the inequalities

01 2

2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5ˆ

00_.0/ ₍₄₎

1

2 2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5

ˆ0 R

f d

ˆ0_.0/ R

f d

Z

ˆıf d ˆ

0

@ Z

f d

1

A

1

2 2

6 4 Z

ˆ0ıf ˆ0.0/ f d

0

@ Z

f d

1

A 2

Z

ˆ0ıf ˆ0.0/

f 1d

3

7 5

1

2 2

6 4 Z

ˆ00ıf

f2d ˆ00.0/ 0

@ Z

f d

1

A 23

(3)

Proof. IfgWI!Ris a differentiable convex function on the interiorIVof the intervalIthen we have thegradient

inequality

g0.t / .t s/g .t / g .s/g0.s/ .t s/ (5)

for anyt; s2 VI :

If we write the inequality (5) for the power functiong .t /Dtr; r1on the interval.0;1/ ;then we have

rtr 1.t s/g .t / g .s/rsr 1.t s/ (6)

for anys; t > 0:

Letn2be a natural number, theng .t /Dtn=2is convex on.0;1/and by takingt Dx2andsDy2then we get from (6) that

n 2x

n 2 x2 y2

xn ynn

2y

n 2

x2 y2

(7)

for anyn2and anyx; y0:

From (7) we have

n 2Œf .u/

n 2 2

6 4Œf .u/

2 0

@ Z

f d

1

A 23

7

5Œf .u/ n

0

@ Z

f d

1

A n

n

2 0

@ Z

f d

1

A n 22

6 4Œf .u/

2 0

@ Z

f d

1

A 23

7 5

for anyn2;or, equivalently

n 2Œf .u/

n n

2Œf .u/ n 2

0

@ Z

f d

1

A 2

Œf .u/n 0

@ Z

f d

1

A n

n

2 0

@ Z

f d

1

A n 22

6 4Œf .u/

2 0

@ Z

f d

1

A 23

7 5 (8)

for anyn2:

Integrating the inequality overuonwe get

n 2 Z

fnd n

2 Z

fn 2d 0

@ Z

f d

1

A 2

Z

fnd

0

@ Z

f d

1

A n

n

2 0

@ Z

f d

1

A n 22

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5 (9)

for anyn2;which is an inequality of interest in itself.

Letm2:If we multiply (9) byan0and sum overnfrom2tomwe get

1 2

Z

m X

nD2 nanfn

!

d 1

2 Z

m X

nD2

nanfn 2 !

d 0

@ Z

f d

1

A 2

(10)

Z

m X

nD2 anfn

!

d m X

nD2 an

0

@ Z

f d

1

A n

1

2 m X

nD2 nan

0

@ Z

f d

1

A n 22

6 4 Z

f2d

0

@ Z

f d

1

A 23

(4)

Observe that

Z

m X

nD0 anfn

!

d m X

nD0 an 0 @ Z f d 1 A n D Z a0d a0 0 @ Z f d 1 A 0 C Z

.a1f / d a1

0 @ Z f d 1 A 1 C Z m X

nD2 anfn

!

d m X

nD2 an 0 @ Z f d 1 A n D Z m X

nD2 anfn

!

d m X

nD2 an 0 @ Z f d 1 A n

for anym2:

From (10) we get

1 2 Z m X nD2 nanfn ! d 1 2 Z m X nD2

nanfn 2 ! d 0 @ Z f d 1 A 2 (11) Z m X

nD0 anfn

!

d m X

nD0 an 0 @ Z f d 1 A n 1 2 m X

nD2 nan 0 @ Z f d 1 A n 22

6 4 Z

f2d

0 @ Z f d 1 A 23 7 5;

for anym2:

Observe that the power seriesP1

nD2nanznand Pm

nD2nanzn 2are convergent onD .0; R/and 1

X

nD2

nanznDz 1 X

nD2

nanzn 1Dz 1 X

nD1

nanzn 1 a1

!

Dz ˆ0.z/ ˆ0.0/

; z2D .0; R/

while

1 X

nD2

nanzn 2D 1 z

1 X

nD2

nanzn 1D ˆ

0_.z/ _ˆ0_.0/

z ; z2D .0; R/X f0g:

Since 0 < f .u/ < Rfor -almost every u in ; then0 < R

f d < R, the series Pm

nD2nanŒf .u/ n_;

Pm

nD2nanŒf .u/ n 2

;Pm

nD0anŒf .u/ n

are convergent for-almost everyuin;P1 nD0an

R

f d

n

and

P1 nD2nan

R

f d

n 2

are convergent and

m X

nD2

nanŒf .u/nDf .u/ ˆ0.f .u// ˆ0.0/ ;

m X

nD2

nanŒf .u/n 2D ˆ

0_{.f .u//} _ˆ0_.0/

f .u/ ;

m X

nD0

anŒf .u/nDˆ .f .u//

for-almost everyuin:

We also have

1 X nD0 an 0 @ Z f d 1 A n Dˆ 0 @ Z f d 1 A and 1 X

nD2 nan 0 @ Z f d 1 A n 2 Dˆ 0 R

f d

ˆ0.0/ R

f d

:

By taking the limit in (11) over m ! 1; interchanging the limit with the integral, we get the third and fourth inequalities in (4).

Sinceˆ0_{is also a convex function on}_{.0; R/}_{then we have by (5)}

ˆ0 0 @ Z f d 1

A ˆ0.0/ˆ00.0/ Z

(5)

and sinceR

f d > 0;we obtain the second inequality in (4). The first inequality is obvious.

By the inequality (5) applied forˆ0_{we also have}

ˆ00.0/ ˆ

0_{.f .u//} _ˆ0_.0/

f .u/ ˆ

00_{.f .u//}

for-almost everyuin:

This implies that

Z

ˆ0ıf ˆ0.0/

f 1dˆ00.0/

and

Z

ˆ0ıf ˆ0.0/

f d

Z

ˆ00ıf f2d;

which prove the fifth inequality in (4).

Remark 2.2. Letˆ .z/DP1

withR > 0orRD 1:Ifxi 2.0; R/andwi 0 .i D1; : : : ; n/withWn WDPn

iD1wi D1;then we have the

inequalities

0 1

2 2

4 n X

iD1 wix_i2

n X

iD1 wixi

!23

5ˆ00.0/ (12)

1

2 2

4 n X

iD1 wix_i2

n X

iD1 wixi

!23

5 ˆ0 Pn

iD1wixi

ˆ0.0/ Pn

iD1wixi

n X

iD1

wiˆ .xi/ ˆ n X

iD1 wixi

!

1

2 2

4 n X

iD1 wixi

ˆ0.xi/ ˆ0.0/ n X

iD1 wi

xi

ˆ0.xi/ ˆ0.0/ n X

iD1 wixi

!23

5

1

2 2

4 n X

iD1

wix_i2ˆ00.xi/ ˆ00.0/ n X

iD1 wixi

!23

5:

We have the following particular inequalities of interest.

Corollary 2.3. Assume thatf W!Ris-measurable and with0 < f .u/for-almost everyuinand such

thatexpıf; .expıf / f; .expıf / f 1₂_{L .; / :}_{Then we have the inequalities}

0 1

2 2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5

1 2

2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5

exp R

f d

1 R

f d

(13)

Z

.expıf / d exp

0

@ Z

f d

1

A

1

2 2

6 4 Z

.expıf 1/ f d 0

@ Z

f d

1

A 2

Z

.expıf 1/ f 1d 3

7 5

1

2 2

6 4 Z

f2.expıf / d 0

@ Z

f d

1

A 23

(6)

The inequality (13) follows by (4) forˆ .z/Dexp.z/DP1 nD0nŠ1z

n_{; z} 2C.

If we use the inequality (4) forˆ .z/D11z D P1

nD0zn; z2D .0; 1/ ;then we can state:

Corollary 2.4. Assume thatf W!Ris-measurable and with0 < f .u/ < 1for-almost everyuinand

such that.1 f / 1; .1 f / 2f; .1 f / 2f 12L .; / :Then we have the inequalities

0

Z

f2d

0

@ Z

f d

1

A 2

(14)

1

2 2

6 4 Z

f2d

0

@ Z

f d

1

A 23

7 5

2 R

f d

1 R

f d

2

Z

.1 f / 1d

0

@1 Z

f d

1

A 1

1

2 2

6 4 Z

.2 f / f2

.1 f /2 d

0

@ Z

f d

1

A 2

Z

2 f

.1 f /2d 3

7 5

Z

f2

.1 f /3d 0

@ Z

f d

1

A 2

:

3 Applications for functions of selfadjoint operators

LetAbe a selfadjoint operator on the complex Hilbert space.H;h:; :i/with the spectrumSp .A/included in the interval Œm; M for some real numbersm < M and letfEg be itsspectral family. Then for any continuous

function f W Œm; M ! R, it is well known that we have the following spectral representation in terms of the

Riemann-Stieltjes integral(see for instance [12, p. 257]):

hf .A/ x; yi D M Z

m 0

f ./ d .hEx; yi/ ; (15)

for anyx; y2H:

The functiongx;y./WD hEx; yiis ofbounded variationon the intervalŒm; M andgx;y.m 0/D0while

gx;y.M /_{D h}x; yifor anyx; y 2H:It is also well known thatgx./_{WD h}Ex; xiismonotonic nondecreasing

andright continuousonŒm; M for anyx2H. The following result holds:

Theorem 3.1. Letˆ .z/DP1

withR > 0orR D 1andAa bonded selfadjoint operator on the Hilbert spaceH withm DminSp .A/and

M DmaxSp .A/ :Assume thatf WI !Ris continuous onIwithŒm; M VIand0 < f .u/ < Rfor anyu2 VI :

Then we have the inequalities

01 2

hD

f2.A/ x; xE hf .A/ x; xi2iˆ00.0/ (16)

1

2 hD

f2.A/ x; xE hf .A/ x; xi2iˆ

0_._h_{f .A/ x; x}_i_/ _ˆ0_.0/

hf .A/ x; xi h.ˆıf / .A/ x; xi ˆ .hf .A/ x; xi/

1

2 ˝

ˆ0ıf

.A/ ˆ0.0/ I

(7)

D

ˆ0ıf

.A/ ˆ0.0/ I

f 1.A/ x; xEhf .A/ x; xi2i

1

2 hD

ˆ00ıf

.A/ f2.A/ x; xE ˆ00.0/ .hf .A/ x; xi/2i

for anyx2H;kxk D1:

In particular, if0 < mM < R;then

0 1

2 hD

A2x; xE hAx; xi2iˆ00.0/ (17)

1

2 hD

A2x; x E

hAx; xi2

iˆ0.hAx; xi/ ˆ0.0/ hAx; xi hˆ .A/ x; xi ˆ .hAx; xi/

1

2 h

˝

ˆ0.A/ ˆ0.0/ I

Ax; x˛ D

ˆ0.A/ ˆ0.0/ I

A 1x; xEhAx; xi2i

1

2 hD

ˆ00.A/ A2x; x E

ˆ00.0/hAx; xi2 i

for anyx2H;kxk D1:

Proof. Letx 2H;kxk D 1:For small" > 0;considerf W Œm "; M !Rcontinuous andg ./D hEx; xi

monotonic nondecreasing onŒm "; M :Utilising the inequality (4) for the positive measuredDdgwe have

0 1

2 2

6 4

M Z

m "

f2./ dhEx; xi 0

@ M Z

m "

f ./ dhEx; xi 1

A 23

7 5ˆ

00_.0/

1

2 2

6 4

M Z

m "

f2./ dhEx; xi 0

@ M Z

m "

f ./ dhEx; xi 1

A 23

7 5

ˆ0RM

m "f ./ dhEx; xi

ˆ0.0/

RM

m "f ./ dhEx; xi

M Z

m "

ˆ .f .// dhEx; xi ˆ 0

@ M Z

m "

f ./ dhEx; xi 1

A

1

2 2

4 M Z

m "

ˆ0.f .// ˆ0.0/

f ./ dhEx; xi

M Z

m "

ˆ0.f .// ˆ0.0/

f 1./ dhEx; xi 0

@ M Z

m "

f ./ dhEx; xi 1

A 23

7 5

1

2 2

6 4

M Z

m "

ˆ00.f .// f2./ dhEx; xi ˆ00.0/ 0

@ M Z

m "

f ./ dhEx; xi 1

A 23

7 5:

Taking the limit over"!0Cwe deduce the desired result (16).

We can give some examples as follows:

Example 3.2. IfA > 0(is a positive definite operator) onH;then we have the exponential inequalities

0 1

2 hD

(8)

1 2

hD

A2x; xE hAx; xi2iexp.hAx; xi/ 1 hAx; xi hexp.A/ x; xi exp.hAx; xi/

1

2 h

h.exp.A/ I / Ax; xi D.exp.A/ I / A 1x; xEhAx; xi2i

1

2 hD

A2exp.A/ x; xE hAx; xi2i

for anyx2H;kxk D1:

Example 3.3. If0 < A < I;then we have

0DA2x; xE hAx; xi2 (19)

1

2 hD

A2x; xE hAx; xi2ih.2I A/ x; xi h.I A/ x; xi2

D.I A/ 1x; xE h.I A/ x; xi 1

1

2 hD

.2I A/ A2.I A/ 2x; xE D.2I A/ .I A/ 2x; xEhAx; xi2i

DA2.I A/ 3x; xE hAx; xi2

for anyx2H;kxk D1:

For recent inequalities for continuous functions of selfadjoint operators see the papers [1], [5], the monographs [10], [11] and the references therein.

Acknowledgement: The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

References

[1] Agarwal R. P., Dragomir S. S., A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces. Comput. Math. Appl. 59 (2010), no. 12, 3785–3812.

[2] Cerone P., Dragomir S. S., A refinement of the Grüss inequality and applications, Tamkang J. Math. 38 (2007), No. 1, 37-49. Preprint RGMIA Res. Rep. Coll., 5 (2) (2002), Art. 14.

[3] Cheng X.-L., Sun J., Note on the perturbed trapezoid inequality, J. Inequal. Pure & Appl. Math., 3(2) (2002), Art. 21.

[4] Dragomir S. S., A Grüss type inequality for isotonic linear functionals and applications. Demonstratio Math. 36 (2003), no. 3, 551– 562. Preprint RGMIA Res. Rep. Coll. 5(2002), Suplement, Art. 12.[Online http://rgmia.org/v5(E).php].

[5] Dragomir S. S., Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces. J. Inequal. Appl. 2010, Art. ID 496821, 15 pp.

[6] Dragomir S. S., Reverses of the Jensen inequality in terms of the first derivative and applications, Acta Math. Vietnam. 38 (2013), no. 3, 429–446. Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 71.[http://rgmia.org/papers/v14/v14a71.pdf]. [7] Dragomir S. S., Some reverses of the Jensen inequality with applications, Bull. Aust. Math. Soc. 87 (2013), no. 2, 177–194.

Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 72.[http://rgmia.org/papers/v14/v14a72.pdf].

[8] Dragomir S. S., A refinement and a divided difference reverse of Jensen’s inequality with applications, Preprint RGMIA Res. Rep. Coll. 14 (2011), Art. 74.[http://rgmia.org/papers/v14/v14a74.pdf].

[9] Dragomir S. S., Ionescu N. M., Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23 (1994), no. 1, 71–78.

[10] Dragomir S. S.,Operator Inequalities of the Jensen, ˇCebyšev and Grüss Type. Springer Briefs in Mathematics. Springer, New York, 2012. xii+121 pp. ISBN: 978-1-4614-1520-6

[11] Dragomir S. S.,Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1

[12] Helmberg G.,Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc. -New York, 1969.