Generalized geometrically convex functions and inequalities

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R E S E A R C H

Open Access

Generalized geometrically convex

functions and inequalities

Muhammad Aslam Noor

1,2*

_{, Khalida Inayat Noor}

2

_{and Farhat Safdar}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics, King} Saud University, Riyadh, Saudi Arabia

2_{Department of Mathematics,} COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

Abstract

In this paper, we introduce and study a new class of generalized functions, called generalized geometrically convex functions. We establish several basic inequalities related to generalized geometrically convex functions. We also derive several new inequalities of the Hermite-Hadamard type for generalized geometrically convex functions. Several special cases are discussed, which can be deduced from our main results.

MSC: 26D15; 26D10; 90C23

Keywords: generalized convex functions; generalized geometrically convex functions; Hermite-Hadamard’s type inequalities; Hölder’s inequality

1 Introduction

During the last few decades, the theory of convex analysis has turned into one of the most interesting and useful ﬁelds to study a wide class of problems arising in pure and applied sciences. Innovative techniques and calculations have yielded diﬀerent directions for the study of convex analysis. In recent years, various inequalities for convex functions and their variant forms have been developed using novel techniques; see [–].

The theory of convex functions is closely related to theory of inequalities. It is well known that a function is convex, if and only if it satisﬁes an integral inequality, which is known as the Hermite-Hadamard inequality; see [, ]. Such types of integral inequalities are useful in ﬁnding the upper and lower bounds. For recent developments and applica-tions, see [–].

The convex sets and convex functions have been extended and generalized in different directions using innovative ideas to study different problems in a general and unified frame work; see [–]. One of the most recent significant generalizations of convex functions is theϕ-convex function, introduced by Gordjiet al.[]. These functions are non-convex functions. For recent developments, see [, –] and the references therein.

The main purpose of this paper is to introduce a new class of generalized convex func-tions, which are called generalized geometrically convex functions. We establish some new results by using the basic inequalities. We derive new Hermite-Hadamard integral inequalities for the generalized geometrically convex functions. Several special cases are considered. Our results are a signiﬁcant and important reﬁnement of well-known results for inequalities.

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2 Preliminaries

LetIbe an interval in real lineR. Letf :I⊂R+= (,∞)→Rbe a diﬀerentiable function on the interiorIofIand letf :I= [x,y]→Randη(·,·) :R×R→Rbe a continuous bifunction. Throughout this paper, we will use the following notation:

R= (–∞, +∞), R+= (,∞) and R–= (–∞, ).

Deﬁnition .([]) LetIbe an interval in real lineR. A functionf :I= [x,y]→Ris said to be generalized convex with respect to an arbitrary bifunctionη(·,·) :R×R→R, if

ftx+ ( –t)y≤f(y) +tηf(x),f(y), ∀x,y∈I,t∈[, ].

Ifη(x,y) =x–y, then the generalized convex function reduces to a convex function. Every convex function is a generalized convex function, but the converse is not true; see, for example, Examples . and ..

Example .([]) For a convex functionf, we may ﬁnd another functionηother than the functionη(x,y) =x–ysuch that f is generalized convex. Considerf(x) =xandη(x,y) = x+y. Then we have

ftx+ ( –t)y=tx+ ( –t)y

≤tx+y+t( –t)xy

≤tx+y+t( –t)x+y

≤y+tx+x+y

=y+tx+y

=f(y) +tηf(x),f(y), ∀x,y∈R,t∈(, ).

Also the fact thatx_≤_y_{+ (}_x₊_y_{) and}_y_≤_y_{, for all}_x_,_y_∈_R_{shows the correctness of} the inequality fort=  andt= , respectively. This means that f is a generalized convex function. Note that the functionf(x) =x_{is generalized convex with respect to all}_η(_x_,_y_{) =}

ax+bywitha≥,b≥– andx,y∈R.

Example .([]) Considerf:R→Ras

f(x) =

⎧ ⎨ ⎩

–x ifx≥,

x ifx< ,

and deﬁne a bifunctionη= –x–yfor allx,y∈R–= (–∞, ). Thenf isη-convex, but the converse is not true.

Deﬁnition .([]) LetI⊂R+. The setIis said to be a geometrically convex set, if

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Deﬁnition .([]) A functionf:I⊂R+= (,∞)→Ris said to be geometrically convex onI, if

fy–txt≤( –t)f(y) +tf(x), ∀x,y∈I,t∈[, ],

where (y–txt) and ( –t)f(y) +t(f(x)) are the weighted geometric mean of two positive numbersxandyand the weighted arithmetic mean off(x) andf(y), respectively.

We now introduce a new class of generalized convex functions on the geometrically convex set with respect to an arbitrary bifunctionη(·,·), which is called the generalized geometrically convex function.

Deﬁnition . A functionf :I⊂R+= (,∞)→Ris said to be generalized geometrically with respect to a bifunctionη(·,·) :R×R→R, if

fy–txt≤( –t)f(y) +tf(y)+ηf(x),f(y), ∀x,y∈I,t∈[, ]. (.) Ifη(x,y) =x–y, then the generalized geometrically convex functions reduce to geomet-rically convex functions given in Deﬁnition ..

Ift=_ in (.), then

f(√xy)≤f(y) + η

f(x),f(y), ∀x,y∈I,t∈[, ], (.) which is called a generalized Jensen geometrically convex function.

We will use the following notations throughout this paper: () arithmetic mean:

A(a,b) =a+b

 ∀a,b∈R+,a=b,

() logarithmic mean:

L(a,b) = b–a

logb–loga ∀a,b∈R+,a=b,

() generalized logarithmic mean:

Ł(a,b) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

[bp+–ap+ (p+)(b–a)]



p _if_p_{= –, ,}

L(a,b) ifp= –, 

e( bb aa)



b–a ifp= .

3 Main results

In this section, we derive some new Hermite-Hadamard type inequalities for generalized geometrically convex functions. We denoteI= [a,b], unless otherwise speciﬁed.

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then

f(√ab) – 

(lnb–lna)

b a



x η

f

ab x

,f(x)

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≤

b a



xf(x) dx≤

f(a) +f(b)  +

 

ηf(a),f(b)+ηf(b),f(a).

Proof Letf be a generalized geometrically convex function onI. Then,∀a,b∈Iandt∈

[, ], we have

fatb–t≤f(b) +tηf(a),f(b). (.) Integrating (.) overton [, ], we have

 

fatb–tdt≤

 

f(b) +tηf(a),f(b)dt

= f(b) +  η

f(a),f(b).

Thus



lnb–lna

b a



x

f(x)dx≤ f(b) + η

f(a),f(b).

Using (.) and takingx= (at_b–t_{) and}_y_{= (}_a–t_bt_{), we have}

f(√ab)≤ fa–tbt+ η

fatb–t,fa–tbt.

Integrating (.) overton [, ], we have

f(√ab) = 

lnb–lna

b a



x f(x) +

 η

f

ab x

,f(x)

dx.

Thus

f(√ab) – 

(lnb–lna)

b a



xη

f

ab x

,f(x)

dx≤ 

lnb–lna

b a



x

f(x)dx.

Sincef is a generalized geometrically convex function,

fatb–t≤f(b) +tηf(a),f(b), (.)

fa–tbt≤f(a) +tηf(b),f(a). (.) Adding (.) and (.), we have

fatb–t+fa–tbt≤f(a) +f(b) +tηf(b),f(a)+tηf(a),f(b). (.) Integrating (.) overton [, ], we have



lnb–lna

b a



x

f(x)dx≤f(a) +f(b) +  

ηf(a),f(b)+ηf(b),f(a).

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Corollary . Ifη(b,a) =b–a in Theorem.,then

f(√ab)≤

b a



xf(x) dx≤

f(a) +f(b)  .

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then



lnb–lna

b a



x

f(x)g(x)dx≤ 

M(a,b) +  N(a,b),

where

M(a,b) =f(b) +ηf(a),f(b)g(b) +ηg(a),g(b)+f(b)g(b), (.)

N(a,b) =f(b)g(b) +ηg(a),g(b)+g(b)f(b) +ηf(a),f(b). (.)

Proof Letf,g be generalized geometrically convex functions onI. Then,∀a,b∈I and

t∈[, ], we have

fatb–t≤( –t)f(b) +tf(b) +ηf(a),f(b), (.)

gatb–t≤( –t)g(b) +tg(b) +ηg(a),g(b). (.)

From (.) and (.), we have

fatb–tgatb–t

≤( –t)f(b)g(b) +t( –t)f(b)g(b) +ηg(a),g(b)+g(b)f(b) +ηf(a),f(b) +tf(b) +ηf(a),f(b)g(b) +ηg(a),g(b). (.)

Integrating both sides of (.) overton [, ], we have

 

fatb–tgatb–tdt

≤f(b)g(b)

 

( –t)dt

+f(b)g(b) +ηg(a),g(b)+g(b)f(b) +ηf(a),f(b) 



t( –t) dt

+f(b) +ηf(a),f(b)g(b) +ηg(a),g(b) 



tdt

=

f(b)g(b) +  

f(b)g(b) +ηg(a),g(b)+g(b)f(b) +ηf(a),f(b) +



f(b) +ηf(a),f(b)g(b) +ηg(a),g(b) =

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Thus



lnb–lna

b a



x

f(x)g(x)dx≤ 

M(a,b) +  N(a,b).

This completes the proof.

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then



lnb–lna

b a



x f(x)g

ab x

dx≤

M(a,b) +  N(a,b),

where M(a,b)and N(a,b)are deﬁned by(.)and(.).

fat_b–t_≤_{( –}_t₎_f₍_b_{) +}_t_f₍_b_{) +}_η_f₍_a_),_f₍_b₎_, _(.)

ga–tbt≤( –t)g(a) +tg(a) +ηg(b),g(a). (.)

From (.) and (.), we have

fatb–tga–tbt

≤( –t)f(b)g(a) +t( –t)f(b)g(a) +ηg(b),g(a) +g(a)f(b) +ηf(a),f(b)

+tf(b) +ηf(a),f(b)g(a) +ηg(b),g(a). (.)

Integrating both sides of (.) overton [, ], we have

 

fatb–tga–tbtdt

≤f(b)g(a)

 

( –t)dt

+f(b)g(a) +ηg(b),g(a)+g(a)f(b) +ηf(a),f(b) 



t( –t) dt

+f(b) +ηf(a),f(b)g(a) +ηg(b),g(a) 



tdt

=

f(b)g(a) +  

f(b)g(a) +ηg(b),g(a)+g(a)f(b) +ηf(a),f(b) +



f(b) +ηf(a),f(b)g(a) +ηg(b),g(a) =

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Thus



lnb–lna

b a



x f(x)g

ab x

dx≤

M(a,b) +  N(a,b).

Corollary . If f =g andη(x,y) =x–y in Theorem.,then



lnb–lna

b a



x f(x)f

ab x

dx≤

f(b)f(a) +  

f(b) +f(a).

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then



lnb–lna

b a



x f(x)g

ab x

dx

≤  

f(b) +g(a)

+g(a) +ηg(b),g(a)+f(b) +ηf(a),f(b) +f(b)f(b) +ηf(a),f(b)+g(a)g(a) +ηg(b),g(a).

fatb–t≤( –t)f(b) +tf(b) +ηf(a),f(b), (.)

ga–tbt≤( –t)g(a) +tg(a) +ηg(b),g(a). (.) Now,



lnb–lna

b a



x f(x)g

ab x

dx

=

 

≤ 

 

fatb–t+ga–tbtdt

≤ 

 

( –t)f(b) +tf(b) +ηf(a),f(b)

+( –t)g(a) +tg(a) +ηg(b),g(a)

dt

= 

f(b) +g(a) 



( –t)dt

+g(a) +ηg(b),g(a)+f(b) +ηf(a),f(b) 



tdt

+f(b)f(b) +ηf(a),f(b)+g(a)g(a) +ηg(b),g(a) 



t( –t) dt

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= 

f(b) +g(a)+g(a) +ηg(b),g(a)+f(b) +ηf(a),f(b) +f(b)f(b) +ηf(a),f(b)+g(a)g(a) +ηg(b),g(a),

which is the required result.

Corollary . If f =g andη(b,a) =b–a in Theorem.,then



lnb–lna

b a



x f(x)f

ab x

dx

≤  

f(b) +f(a)+f(b)f(a) +f(a)f(b).

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then



lnb–lna

b a



x f(x)g

ab x

dx

=  

f(b) +g(a)+f(b) +ηf(a),f(b)g(a) +ηg(b),g(a) +f(b) +g(a)f(b) +ηf(a),f(b)g(a) +ηg(b),g(a).

fatb–t≤( –t)f(b) +tf(b) +ηf(a),f(b), (.)

ga–tbt≤( –t)g(a) +tg(a) +ηg(b),g(a). (.) Now,



lnb–lna

b a



x f(x)g

ab x

dx

=

 

≤  

 

fatb–t+ga–tbtdt

≤  

 

[( –t)f(b) +tf(b) +ηf(a),f(b)

+( –t)g(a) +tg(a) +ηg(b),g(a)

dt

= 

 

( –t)f(b) +g(a)+tf(b) +ηf(a),f(b)g(a) +ηg(b),g(a)

dt

= 

 

[( –t)f(b) +g(a)+tf(b) +ηf(a),f(b)g(a) +ηg(b),g(a)

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=  

f(b) +g(a)+f(b) +ηf(a),f(b)g(a) +ηg(b),g(a) +f(b) +g(a)f(b) +ηf(a),f(b)g(a) +ηg(b),g(a),

Corollary . If f =g andη(b,a) =b–a in Theorem.,then



lnb–lna

b a



x f(x)f

ab x

dx

≤  

f(b) +f(a)+f(a)f(b)+f(a)f(b)f(b) +f(a).

Theorem . Let f,g:I= [a,b]→(,∞)be generalized geometrically convex functions on I and a,b∈I with a<b.Then

f(√ab)g(√ab)

≤ 

lnb–lna

b a



x

f(x)g(x)dx

+  (lnb–lna)

b a



x

f(x) η

g

ab x

,g(x)

+g(x) η

f

ab x

,f(x)

dx

+  (lnb–lna)

b a



x η(f

ab x

,f(x) η

g

ab x

,g(x)

dx.

f(√ab)≤fa–tbt+ η

fatb–t,fa–tbt, (.)

g(√ab)≤ga–tbt+ η

gatb–t,ga–tbt. (.) From (.) and (.), we have

f(√ab)g(√ab)

≤ fa–tbt+ η

fatb–t,fa–tbt ga–tbt+ η

gatb–t,ga–tbt

=fa–tbtga–tbt+ 

fa–tbtηgatb–t,ga–tbt

+ga–tbtηfatb–t,fa–tbt

+ 

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Integrating (.) overton [, ], we have

 

f(√ab)g(√ab) dt

≤  

fa–tbtga–tbtdt

+ 

 

fa–tbtηgatb–t,ga–tbt

+ga–tbtηfatb–t,fa–tbtdt

+ 

 

ηgatb–t,ga–tbtηfatb–t,fa–tbtdt.

Thus

f(√ab)g(√ab)

≤ 

lnb–lna

b a



x

f(x)g(x)dx

+  (lnb–lna)

b a



x

f(x) η

g

ab x

,g(x)

+g(x) η

f

ab x

,f(x)

dx

+  (lnb–lna)

b a



x η

f

ab x

,f(x) η

g

ab x

,g(x)

dx.

Corollary . If f =g andη(b,a) =b–a in Theorem.,then

f(√ab)f(√ab)≤ 

lnb–lna

b a



x

f(x)f(x)dx.

Theorem . Let f,g:I= [a,b]→(,∞)be increasing and generalized geometrically con-vex functions on I and a,b∈I with a<b.Then



lnb–lna

b a



x

f(x) g(b) +  η

g(a),g(b)dx

+ 

lnb–lna

b a



x

g(x) f(b) + η

f(a),f(b)dx

≤ 

lnb–lna

b a



x

f(x)g(x)dx

+ f(b) + η

f(a),f(b) g(b) +  η

g(a),g(b).

Proof Letf andgbe generalized geometrically convex functions onI. Then∀a,b∈Iand

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and

gatb–t≤g(b) +tηg(a),g(b). (.)

Using the inequality

(a–b)(c–d)≥ ∀a,b,c,d∈R,a<b,c<d, (.)

we have

fatb–tg(b) +tηg(a),g(b)+gatb–tf(b) +tηf(a),f(b) ≤fatb–tgatb–t+f(b) +tηf(a),f(b)g(b) +tηg(a),g(b).

Integrating the above inequality overton [, ], we have

 

fatb–tdt

 

g(b) +tηg(a),g(b)dt

+

 

gatb–tdt

 

≤

 

fatb–tdt

 

gatb–tdt

+

 

g(b) +tηg(a),g(b)dt.

Now after simple integration, we have



lnb–lna

b a



x

f(x) g(b) + η

g(a),g(b)dx

+ 

lnb–lna

b a



x

g(x) f(b) +  η

f(a),f(b)dx

≤ 

lnb–lna

b a



x

f(x)g(x)dx

+ f(b) + η

f(a),f(b) g(b) +  η

g(a),g(b).

Corollary . If f =g andη(b,a) =b–a in Theorem.,then



lnb–lna

b a



x

f(x) f(b) +f(a) 

dx

≤ 

lnb–lna

b a



x

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4 Integral inequalities

In this section, we will use the following result to obtain our main results.

Lemma .([]) Let f:I⊂R+= (,∞)→Rbe a diﬀerentiable function on the interior

I_{of I}_,_{where a}_,_b_∈_{I with a}_<_{b and f}_∈_L_[_a_,_b_]._Then

bf(b) –af(a) –

b a

f(x) dx

=(lnb–lna) 





b+ta–tfb+t_a–t_d_t₊

 

b–ta+tfb–t_a+t_d_t

.

Theorem . Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on the interior I

of I,where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q≥,then

bf(b) –af(a) –

b a

f(x) dx

≤(b–a) –_q

()q+

bb–a–L(a,b)f(b)q

+L(a,b) –af(b) +ηf(a),f(b)q  q

+ab– a+L(a,b)f(b) +ηf(a),f(b)q+b–L(a,b)f(b)qq_.

Proof Using Lemma . and Hölder’s inequality, we have

bf(b) –af(a) –

b a

f(x) dx

=ab(lnb–lna)    b a t

fb+t_a–t_d_t₊

  a b t

fb–t_a+t_d_t

≤ab(lnb–lna)    b a t dt

–_q  

b a

t

fb+t_a–tq_d_t

 q +   a b t dt

–_q  

a b

t

_f_b–t

 _a+tq_d_t

 q

. (.)

Now consider

I =

  b a t

≤

 

_f₍_b₎q +t 

+f(b) +ηf(a),f(b)q

 –t



dt

=f(b)q

  a b t  +t



dt+f(b) +ηf(a),f(b)q

  b a t  –t



dt

=f(b)qb–a–L(a,b) a(lnb–lna) +f

₍_b_{) +}_η_f₍_a_),_f₍_b₎q L(a,b) –a

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SimilarlyIbecomes

 

a b

t

fb–t_a+tq_d_t

≤f(b) +ηf(a),f(b)qb– a+L(a,b) b(lnb–lna) +f

₍_b₎q b–L(a,b)

b(lnb–lna). (.)

Also





b a

t dt

–_q =

b–a a(lnb–lna)

–_q

. (.)

Combining (.), (.), (.) and (.), we have

bf(b) –af(a) –

b a

f(x) dx

≤(b–a) –q

()q+

bb–a–L(a,b)f(b)q

+L(a,b) –af(b) +ηf(a),f(b)qq

+ab– a+L(a,b)f(b) +ηf(a),f(b)q+b–L(a,b)f(b)qq_,

Ifη(b,a) =b–a, then Theorem . reduces to the following result.

Corollary .([]) Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior

of I),where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q≥,then

bf(b) –af(a) –

b a

f(x) dx

≤(b–a) –_q

()q+

bb–a–L(a,b)f(b)q+L(a,b) –af(a)q  q

+ab– a+L(a,b)f(a)q+b–L(a,b)f(b)q  q_.

Theorem . Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior of

I),where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q> ,then

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) ()+q

L(a

q q–_,_b

q q––



q_b_A__f₍_b₎q_,_f₍_b_{) +}_η_f₍_a_),_f₍_b₎qq

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bf(b) –af(a) – 

b–a

b a

f(x) dx

_f_b+t

 _a–t_d_t₊

  a b t

_f_b–t

 _a+t_d_t

≤ab(lnb–lna)    b a qt q– dt

–_q  

_f_b+t

 _a–tq_d_t

 q +   a b qt q– dt

–_q  

 q

. (.)

Now we consider

I =

 

≤

 

f(b)q

 +t



+f(b) +ηf(a),f(b)q

 –t



dt

=f(b)q

 

 +t



 

 –t



dt

=  f

₍_b₎q +

f

₍_b_{) +}_η_f₍_a_),_f₍_b₎q

. (.)

 

fb–t_a+tq_d_t≤ f

₍_b_{) +}_η_f₍_a_),_f₍_b₎q + 

f

₍_b₎q

. (.) Also   b a qt q–

dt]–q₌

bqq–_–_a q q–

(_qq_–)a

q

q–₍_ln_b_–_ln_a₎

–_q

. (.)

Combining (.), (.), (.) and (.), we have

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) ()+q

Laqq–_,_b q q––



q_b_A__f₍_b₎q_,_f₍_b_{) +}_η_f₍_a_),_f₍_b₎qq

+aAf(b) +ηf(a),f(b)q,f(b)qq_,

Ifη(b,a) =b–a, then Theorem . reduces to the following result.

Corollary .([]) Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior

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function on[a,b]for q> ,then

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) ()+q

La

q q–_,_b

q q––



q_b_A__f₍_b₎q_,_f₍_a₎qq

+aAf(a)q,f(b)qq_.

Theorem . Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior of

I),where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q≥,then

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) –_q

(q)q

bLaq,bq–aqf(b) +ηf(a),f(b)q

+bq–aq–Laq,bqf(b)qq

+af(b) +ηf(a),f(b)qbq– aq+Laq,bq

+bq–Laq,bqf(b)q_.

bf(b) –af(a) –

b a

f(x) dx

=ab(lnb–lna) 





b a

t

fb+t_a–t_d_t₊

 

a b

t

fb–t_a+t_d_t

≤ab(lnb–lna) 



  dt

–_q  

b a

qt

 q

+ 

  dt

–_q  

a b

qt

 q

. (.)

Now consider

I =

 

b a

qt

≤

 

b a

qt

f(b)t

 +t



+f(b) +ηf(a),f(b)q

 –t



dt

=f(b)q

 

b a

qt  +t



 

b a

qt  –t



dt

=f(b)qb

q_–_aq_–_L₍_aq_,_bq₎

qaq₍_ln_b_–_ln_a₎ +f

₍_b_{) +}_η_f₍_a_),_f₍_b₎q L(aq,bq) –aq

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 

a b

qt

≤f(b) +ηf(a),f(b)qb

q_{– }_aq₊_L₍_aq_,_bq₎

qbq₍_ln_b_–_ln_a₎ +f

₍_b₎q bq–L(aq,bq)

qbq₍_ln_b_–_ln_a₎. (.)

Combining (.), (.) and (.), we have

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna)–  q

(q)q

bLaq,bq–aqf(b) +ηf(a),f(b)q

+bq–aq–Laq,bqf(b)q  q

+af(b) +ηf(a),f(b)qbq– aq+Laq,bq+bq–Laq,bqf(b)  q_,

Ifη(b,a) =b–a, then Theorem . reduces to the following result.

Corollary .([]) Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior

of I),where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q≥,then

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) –_q

(q)q

bLaq,bq–aqf(a)q

+bq–aq–Laq,bqf(b)qq₊_a_f₍_a₎q_bq_{– }_aq₊_L_aq_,_bq +bq–Laq,bqf(b)q_.

Theorem . Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on I(interior of

I),where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q> and q>p> ,then

bf(b) –af(a) –

b a

f(x) dx

≤(lnb–lna) (p)q

La

q–p q–_,_b

q–p q––



q_a–pq_b_L_ap_,_bp_–_ap_f₍_b_{) +}_η_f₍_a_),_f₍_b₎q

+bp–ap–Lap,bpf(b)q  q +ab–

p

q_bp_{– }_ap₊_L_ap_,_bp_f₍_b_{) +}_η_f₍_a_),_f₍_b₎q +bq–Lap,bpf(b)q

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bf(b) –af(a) –

b a

f(x) dx

fb+t_a–t_d_t₊

  a b t

fb–t_a+t_d_t

≤ab(lnb–lna)    b a

(q–p)t q–

dt

–_q  

b a

pt

 q +   a b

(q–p)t q–

dt

–_q  

a b

pt

 q

. (.)

Now consider

I =

  b a pt

≤   b a pt

_f₍_b₎t +t 

+f(b) +ηf(a),f(b)q

 –t



dt

=f(b)q

  a b pt  +t



  b a pt  –t



dt

=f(b)qb

p_–_ap_–_L₍_ap_,_bp₎

pap₍_ln_b_–_ln_a₎ +f

₍_b_{) +}_η_f₍_a_),_f₍_b₎q L(ap,bp) –ap

pap₍_ln_b_–_ln_a₎. (.)

  a b pt

≤f(b) +ηf(a),f(b)qb

p_{– }_ap₊_L₍_ap_,_bp₎

pbp₍_ln_b_–_ln_a₎ +f

₍_b₎q bp–L(ap,bp)

pbp₍_ln_b_–_ln_a₎. (.)

Also   b a

(q–p)t q–

dt

–q =

b

q–p q–_–_a

q–p q–

(q_q–_–p)aqq––p₍_ln_b_–_ln_a₎

–q

. (.)

Combining (.), (.), (.) and (.), we have

bf(b) –af(a) –

b a

f(x) dx

La

q–p q–_,_b

q–p q––



q_a–pq_b_L_ap_,_bp_–_ap_f₍_b_{) +}_η_f₍_a_),_f₍_b₎q

+ (bp_–_ap_–_L_ap_,_bp_f₍_b₎qq +ab–

p

q_bp_{– }_ap₊_L_ap_,_bp_f₍_b_{) +}_η_f₍_a_),_f₍_b₎q +bq–Lap,bpf(b)qq_,

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Ifη(b,a) =b–a, then Theorem . reduces to the following result.

Corollary .([]) Let f :I⊂R+= (,∞)→Rbe a diﬀerentiable function on the

inte-rior Iof I,where a,b∈I with a<b and f∈L[a,b].If|f|is a generalized geometrically convex function on[a,b]for q> and q>p> ,then

bf(b) –af(a) –

b a

f(x) dx

La

q–p q–_,_b

q–p q––



q_a–pq_b_L_ap_,_bp_–_ap_f₍_a₎q

+bp–ap–Lap,bpf(b)q 

q₊_ab–pq_bp_{– }_ap₊_L_ap_,_bp_f₍_a₎q +bq–Lap,bpf(b)qq_.

5 Conclusion

In this paper, we have introduced and investigated a new class of generalized functions, called generalized geometrically convex functions. Some basic inequalities related to gen-eralized geometrically convex functions have been derived. Several new inequalities of the Hermite-Hadamard type for generalized geometrically convex functions have been estab-lished. Several special cases are discussed, which can be deduced from our main results. The techniques and ideas of this paper may stimulate further research in this dynamic ﬁeld. It is an interesting problem to consider these estimates in the setting of fractional calculus of meromorphic functions; see [, , , , ] and the references therein.

Acknowledgements

The authors would like to thank the Rector of the COMSATS Institute of Information Technology, Pakistan, for providing excellent research and academic environments. The authors are pleased to acknowledge the support of the Distinguished Scientist Fellowship Program (DSFP), from King Saud University, Riyadh, Saudi Arabia. The authors are grateful to the referees for their valuable and constructive comments.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

MAN, KIN and FS worked jointly. All the authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 11 July 2017 Accepted: 17 August 2017

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