Fractional Hermite Hadamard inequalities containing generalized Mittag Leffler function

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R E S E A R C H

Open Access

Fractional Hermite-Hadamard inequalities

containing generalized Mittag-Lefﬂer

function

Marcela V Mihai

1

_{, Muhammad Uzair Awan}

2*

_{, Muhammad Aslam Noor}

3

_{and Khalida Inayat Noor}

3

*_{Correspondence:}

[email protected]

2_{Department of Mathematics, GC}

University, Faisalabad, Pakistan Full list of author information is available at the end of the article

Abstract

The objective of this paper is to establish some new reﬁnements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leﬄer function.

MSC: 26D15; 26A51; 26A33; 33E12

Keywords: harmonic convex function; Hermite-Hadamard inequalities; Mittag-Leﬄer function

1 Introduction

A functionf :I→Ris said to be convex if

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

Convexity plays a pivotal role in different fields of pure and applied sciences. Another fact that makes it more attractive is its close relationship with theory of inequalities. In particular, integral inequalities have been obtained via convex functions. Inspired by the research work in this field, many authors introduced new extensions of classical convex functions; see, for example, [–] and the references therein. Recently, Işcan [] intro-duced and investigated the notion of harmonically convex functions. These days the class of harmonically convex functions is receiving much attention by many researchers. For more details, see [–]. Hermite and Hadamard independently obtained an integral in-equality that provides us a necessary and sufficient condition for a function to be convex. This famous result reads as follows.

Letf :I⊇[a,b]→Rbe a convex function, then

f

a+b 

≤  b–a

b a

f(x) dx≤f(a) +f(b)

 . (.)

For more details on Hermite-Hadamard-type inequalities, see [, , , , ]. Sarikaya et al. [] developed a new generalization of Hermite-Hadamard-type inequality using fractional calculus approach. This opened a new venue of research in this ﬁeld. Utilizing

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the concepts of fractional calculus, Işcan et al. [] derived new refinements of fractional Hermite-Hadamard’s inequality via harmonically convex functions. The main motivation of this paper is to obtain new refinements of fractional Hermite-Hadamard-type inequal-ities via harmonically convex functions in connection with the generalized Mittag-Leffler function, which even generalizes the classical Riemann-Liouville fractional integral oper-ators. We also discuss some particular cases.

2 Preliminaries

In this section, we discuss some preliminary concepts and facts. Recently, Işcan [] ob-tained several inequalities of Hermite-Hadamard type via harmonic convex functions.

The class of harmonic convex functions is deﬁned as follows.

Deﬁnition .([]) Letf:I⊆R\ {} →R, whereIa real interval. The functionf is said to be harmonic convex if

f

xy tx+ ( –t)y

≤tf(y) + ( –t)f(x) (.)

for allx,y∈Iandt∈[, ]. If (.) holds in the reversed sense, thenfis said to be harmonic concave.

Işcan et al. [] established new fractional estimates of Hermite-Hadamard-type inequal-ities via harmonic convex functions. For more details on Hermite-Hadamard inequalinequal-ities involving fractional integrals; see Mihai [, ], Mihai et al. [], Awan et al. [], Kunt et al. [], Sarikaya et al. [], Nisan et al. [], and references therein. Latif et al. [] gave the following deﬁnition.

Deﬁnition . A functiong: [a,b]⊆R\ {} →Ris said to be harmonically symmetric with respect to _a₊ab_b if

g(x) =g

  a+

 b–

 x

for allx∈[a,b].

For details and the definition of Riemann-Liouville fractional integrals, see [, ]. Salim et al. [] have defined the generalized fractional integral operators containing Mittag-Leffler function:

Deﬁnition . Letμ,ν,k,l,γ >  andω∈R. Then the generalized fractional integral op-erators containing the Mittag-Leﬄer functions ε_μγ,_,_νδ,_,k_l_,_ω_,_a+ andε

γ,δ,k

μ,ν,l,ω,b– for a real-valued continuous functionf are deﬁned by

ε_μγ,_,_νδ,_,k_l_,_ω_,_a+f

(x) =

x a

(x–t)ν–E_μγ_,,_νδ,_,k_lω(x–t)μf(t) dt (.)

and

ε_μγ,_,_νδ,_,k_l_,_ω_,_b–f

(x) =

b x

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respectively, whereEγ_μ,_,δ_ν,_,k_l is the generalized Mittag-Leﬄer function deﬁned as

Eγμ,,δν,,kl(t) = ∞

n= (γ)kn

(μn+ν) tn (δ)ln

,

and (a)nis the Pochhammer symbol deﬁned as

(a)n=a(a+ )· · ·(a+n– ), (a)= .

Remark Ifk=l=  in (.), then the integral operator (εγ_μ,_,δ_ν,_,,k_ω_,_a+f) reduces to the integral operator (ε_μγ,_,δ_ν,_,_l_,_ω_,_a+f) containing the generalized Mittag-Leffler functionEμγ,,δν,,introduced by Srivastava and Tomovski []. Ifk=l=  andδ= , then (.) reduces to the integral operator defined by Prabhaker [] and containing the Mittag-Leffler functionEγμ,ν. For

ω=  in (.), the integral operator (ε_μγ,_,_νδ,_,k_l_,_ω_,_a+f) reduces to the Riemann-Liouville fractional integral operator [].

In [] the properties of the generalized integral operator and the generalized Mittag-Leﬄer function are studied. It is proved thatE_μγ,_,δ_ν,_,k_l(t) is absolutely convergent for allt∈R ifk<l+μ. Since

E_μγ_,,δ_ν,_,k_l(t)≤ ∞

n=

(γ)kn

(μn+ν) tn (δ)ln

with

∞

n=

(γ)kn

(μn+ν) tn (δ)ln

=S,

we have

_Eγ,δ,k

μ,ν,l(t)≤S.

3 Results and discussions

In this section, we discuss our main results. We write g ∞=sup_t_∈[_a_,_b_]|g(t)|for a contin-uous functiong: [a,b]→R.

Lemma . If g: [a,b]⊆R\ {} →Ris integrable and harmonically symmetric with re-spect to _a₊ab_b,then

εγ,δ,k

μ,ν,l,ω,(_a+_abb)+g◦h

 a

=εγ,δ,k

μ,ν,l,ω,(_a_ab+b)–g◦h

 b

=

 ε

γ,δ,k

μ,ν,l,ω,(_a_ab+b)+g◦h

 a

+εγ,δ,k

μ,ν,l,ω,(_a_ab+b)–g◦h

 b

,

where h(x) =  x,x∈[

 b,

 a] .

Proof Sincegis harmonically symmetric with respect to_bab_–_a, using Deﬁnition ., we have g(_t) =g( 

a+b–t

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du= –dtgives

εγ,δ,k

 a

=



a

a+b

ab

 a–t

ν– E_μγ,_,δ_ν,_,k_l

ω

 a–t

μ

g

 t

dt

=



a

a+b

ab

 a–t

ν– E_μγ,_,δ_ν,_,k_l

ω

 a–t

μ

g

  a+

 b–t

dt

=

a+b

ab



b

u– b

ν– Eγ_μ,_,δ_ν,_,k_l

ω

u– b

μ

g

 u

du=εγ,δ,k

μ,ν,l,ω,(a__ab+b)–g◦h

 b

.

This completes the proof.

Lemma . Let f :I⊂(, +∞)→Rbe a diﬀerentiable function on I◦,the interior of I,such that f∈L[a,b],where a,b∈I.If g: [a,b]→Ris integrable and harmonically symmetric with respect to_aab₊_b,then the following equality holds:

f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)–g◦h

 b

– εγ,δ,k

μ,ν,l,ω,(_a+_abb)+fg◦h

 a

+εγ,δ,k

μ,ν,l,ω,(_a+_abb)–fg◦h

 b

=

a+b

ab



b

t 

b

s– b

ν– E_μγ_,,δ_ν,_,k_l

ω

s– b

μ

(g◦h)(s) ds

(f◦h)(t) dt

–



a

a+b

ab



a

t

 a–s

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–s

μ

(g◦h)(s) ds

(f ◦h)(t) dt, (.)

where h(x) =_x,x∈[_b,_a],andω= (_bab_–_a)μ_ω_.

Proof It suﬃces to show that

I=

a+b

ab



b

t 

b

s– b

ν– Eμγ,,δν,,kl

ω

s– b

μ

(g◦h)(s) ds

(f◦h)(t) dt

–



a

a+b

ab



a

t

 a–s

ν– Eγ_μ,_,_νδ,_,k_l

ω

 a–s

μ

(g◦h)(s) ds

(f◦h)(t) dt

=I–I. (.)

By Lemma ., integrating by parts, we have

I=

t 

b

s– b

ν– Eγ_μ,_,δ_ν,_,k_l

ω

s–  b

μ

(g◦h)(s) ds·(f◦h)(t)

a+b

ab



b

–

a+b

ab



b

t– b

ν– E_μγ_,,δ_ν,_,k_l

ω

t– b

μ

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=f

ab a+b



b

s– b

ν– E_μγ,_,δ_ν,_,k_l

ω

s–  b

μ

(g◦h)(s) ds

–

a+b

ab



b

t– b

ν– E_μγ_,,δ_ν,_,k_l

ω

t– b

μ

(fg◦h)(t) dt. (.)

Thus

I=f

ab a+b

εγ,δ,k

 b

–εγ,δ,k

μ,ν,l,ω,(a_+_abb)–fg◦h

 b

= f

ab a+b ε

γ,δ,k

μ,ν,l,ω,(a_+_abb)+g◦h

 a

+εγ,δ,k

 b

–εγ,δ,k

 b

. (.)

Analogously,

I=



a

t

 a–s

ν– E_μγ,_,δ_ν,_,k_l

ω

 a–s

μ

(g◦h)(s) ds·(f◦h)(t) 

a

a+b

ab

–



a

a+b

ab

 a–t

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–t

μ

(g◦h)(t)(f ◦h)(t) dt

= –f

ab a+b



a

a+b

ab

 a–s

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–s

μ

(g◦h)(s) ds

+



a

a+b

ab

 a–t

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–t

μ

(fg◦h)(t) dt. (.)

Also,

I= –f

ab a+b

εγ,δ,k

 a

+εγ,δ,k

 a

= – f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

μ,ν,l,ω,(_a+_abb)–g◦h

 b

+εγ,δ,k

 a

. (.)

Inserting (.) and (.) into (.), we get (.), and the proof is complete.

Remark Takingω=  in Lemmas . and ., we have Lemmas  and  from [].

The next result is an Hermite-Hadamard-type inequality for a generalized fractional integral operator containing the generalized Mittag-Leﬄer function.

Theorem . Let f :I⊂(, +∞)→Rbe a function such that f ∈L[a,b],where a,b∈I.

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integrals holds:

f

ab a+b

εγ,δ,k

μ,ν,l,ω,(a_+_abb)–

 b

≤   ε

γ,δ,k

μ,ν,l,ω,(_a+_abb)–f ◦h

 b

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)+f ◦h

 a

≤f(a) +f(b) 

εγ,δ,k

μ,ν,l,ω,(a_+_abb)+

 a

, (.)

where h(x) =_x,x∈[_b,_a],andω= (_bab_–_a)μ_ω_.

Proof Sincef is a harmonically convex function on [a,b], we have

f

xy x+y

≤f(x) +f(y)

 (.)

forx,y∈[a,b]. Substitutingx= t ab

a+–tb

andy= t ab

b+–ta

into inequality (.), we get

f

ab a+b

≤f( ab

t

a+–tb

) +f(t ab

b+–ta )

 . (.)

Multiplying both sides of (.) bytν–_Eγ,δ,k

μ,ν,l(ωtμ) and integrating over [, ], we get

f

ab a+b





tν–E_μγ_,,δ_ν,_,k_lωtμdt

≤

 

tν–E_μγ_,,_νδ,_,k_lωtμf

ab t a+

–t  b

dt+

 

tν–E_μγ_,,δ_ν,_,k_lωtμf

ab t b+

–t  a

dt.

Setting

u=  ab

t a+

 –t  b

, v= 

ab

t b+

 –t  a

, (.)

we have

f

ab a+b

a+b

ab



b

v– b

ν– Eγμ,,δν,,kl

ω

v– b

μ

dv

≤



a

a+b

ab

 a–u

ν– E_μγ,_,δ_ν,_,k_l

ω

 a–u

μ

(f◦h)(u) du

+

a+b

ab



b

v– b

ν– Eγ_μ,_,_νδ,_,k_l

ω

v– b

μ

(f ◦h)(v) dv.

This implies

f

ab a+b

εγ,δ,k

μ,ν,l,ω,(a_+_abb)–

 b

≤   ε

γ,δ,k

 b

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)+f ◦h

 a

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On the other hand, the harmonic convexity off yields

f

ab t a+

–t  b

+f

ab t b+

–t  a

≤f(a) +f(b) (.)

for allt∈[, ]. Multiplying both sides of (.) bytν–_Eγ,δ,k

μ,ν,l(ωtμ) and integrating over [, ], we have

 

ab t a+

–t  b

dt+

 

ab t b+

–t  a

dt

≤f(a) +f(b) 



tν–Eγμ,,δν,,kl

ωtμdt.

This implies, using substitutions (.),



a

a+b

ab

 a–u

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–u

μ

(f ◦h)(u) du

+

a+b

ab



b

v– b

ν– Eγμ,,νδ,,kl

ω

v– b

μ

(f ◦h)(v) dv.

≤f(a) +f(b) 

a

a+b

ab

 a–u

ν– E_μγ_,,δ_ν,_,k_l

ω

 a–u

μ

(f◦h)(u) du.

So

  ε

γ,δ,k

 b

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)+f◦h

 a

≤f(a) +f(b) 

εγ,δ,k

μ,ν,l,ω,(a_+_abb)+

 a

. (.)

Combining (.) and (.), we get (.).

Remark In Theorem ., if we takeω= , then we get the known inequality of Işan et al. []

f

ab a+b

≤(ν+ )

–ν Ja_+_abb+(f◦h)

 a

+Ja+b

ab–(f◦h)

 b

≤f(a) +f(b)

 .

Theorem . Let f :I⊂(, +∞)→Rbe a harmonically convex functsuch that f ∈L[a,b]. If g: [a,b]→Ris nonnegative,integrable,and harmonically symmetric with respect to_aab₊_b, then the following inequality holds:

f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

 b

≤  ε

γ,δ,k

 a

+εγ,δ,k

 b

≤f(a) +f(b)

 ε

γ,δ,k

 a

+εγ,δ,k

 b

, (.)

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Proof Sincef ia a harmonically convex function on [a,b], multiplying both sides of (.) by tν–_Eγ,δ,k

μ,ν,l(ωtμ)g(t ab

a+–tb

) and then integrating the resulting inequality over [, ], we obtain

f

ab a+b





tν–E_μγ_,,δ_ν,_,k_lωtμg

ab t a+

–t  b dt ≤  

tν–_Eγ,δ,k

μ,ν,l

ωtμ_f

ab t a+

–t  b g ab t a+

–t  b dt +  

tν–Eγμ,,δν,,kl

ωtμf

ab t b+

–t  a g ab t a+

–t  b

dt.

Sincegis harmonically symmetric with respect to_a₊ab_b, using Deﬁnition ., we haveg(_x) = g( 

a+b–x

) for allx∈[_b,_a]. Settingx=_ab (_ta+–_tb) gives

f

ab a+b



a

a+b

ab

 a–x

ν– Eγ_μ,_,δ_ν,_,k_l

ω

 a–x

μ g  x dx ≤   a

a+b

ab

 a–x

ν– E_μγ_,,δ_ν,_,k_l

ω

 a–x

μ f  x g  x dx +  a

a+b

ab

 a–x

ν– Eμγ,,δν,,kl

ω

 a–x

μ f   a+  b–x

g  x dx .

Using substitutionu=_a+_b–x, we have

f

ab a+b



a

a+b

ab

 a–x

ν– Eγμ,,δν,,kl

ω

 a–x

μ g  x dx ≤   a

a+b

ab

 a–x

ν– E_μγ_,,δ_ν,_,k_l

ω

 a–x

μ f  x g  x dx +

a+b

ab



b

u– b

ν– Eγ_μ,_,δ_ν,_,k_l

ω

u– b μ f  u g  u dx .

Hence, using Lemma ., we obtain

f

ab a+b ε

γ,δ,k

μ,ν,l,ω,(a__ab+b)+g◦h

 a

+εγ,δ,k

μ,ν,l,ω,(a__ab+b)–g◦h

 b

≤  ε

γ,δ,k

 a

+εγ,δ,k

 b

. (.)

For the proof of the second inequality in (.), we ﬁrst note thatfis a harmonically convex function. Then, multiplying both sides of (.) by tν–_Eγ,δ,k

μ,ν,l(ωt

μ_)g( ab

t

a+–tb

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integrat-ing the resultintegrat-ing inequality over [, ], we obtain

 

tν–_Eγ,δ,k

μ,ν,l

ωtμ_f

ab t a+

–t  b

g

ab t a+

–t  b

dt

+

 

ab t b+

–t  a

g

ab t a+

–t  b

dt

≤f(a) +f(b) 



tν–E_μγ_,,δ_ν,_,k_lωtμg

ab t a+

–t  b

dt.

From this, using Lemma ., we get

  ε

γ,δ,k

 a

+εγ,δ,k

 b

≤f(a) +f(b)

 ε

γ,δ,k

 a

+εγ,δ,k

 b

.

(.)

From (.) and (.) we obtain (.). The proof is completed.

Theorem . Let f :I⊂(, +∞)→Rbe a diﬀerentiable function such that f∈L[a,b], where a,b∈I,a<b.If|f|is a harmonically convex function and g: [a,b]→Ris continu-ous and harmonically symmetric with respect to _a₊ab_b,then the following inequality holds:

f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

 b

– εγ,δ,k

 a

+εγ,δ,k

 b

≤f

ab a+b

g ∞ εγ,δ,k

μ,ν,l,ω,(a_+_abb)+

 a

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)–

 b

+S g ∞

b–a ab

ν

· ν+ 

ν(ν+ )f(a)+f(b), (.)

where h(x) =_x,x∈[_b,_a],ω= (_bab_–_a)μ_ω_,_and_|_Eγ,δ,k

μ,ν,l(t)| ≤S.

Proof From Lemma ., relationships (.) and (.), and the property of the modulus we have

J=f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

 b

– εγ,δ,k

 a

+εγ,δ,k

 b

≤f

ab a+b

aab+b



b

s–  b

ν–

Eγμ,,νδ,,kl

ω

s– b

μ

(g◦h)(s)ds

+

a+b

ab



b

t– b

ν–

Eγ_μ,_,δ_ν,_,k_l

ω

t– b

μ

(10)

+f

ab a+b

a

a+b

ab

 a–s

ν–

Eμγ,,δν,,kl

ω

 a–s

μ

(g◦h)(s)ds

+



a

a+b

ab

 a–t

ν–

E_μγ_,,δ_ν,_,k_l

ω

 a–t

μ

(fg◦h)(t)dt.

Since g ∞=sup_t_∈[_a_,_b_]|g(t)|and|Eγ_μ,_,δ_ν,_,k_l(t)| ≤S, we have

J≤f

ab a+b

g ∞

ab a+b



b

s– b

ν– Eγ_μ,_,_νδ,_,k_l

ω

s– b

μ

ds

+S g ∞

a+b

ab



b

t– b

ν–

(f◦h)(t)dt

+f

ab a+b

g ∞



a

a+b

ab

 a–s

ν– E_μγ_,,δ_ν,_,k_l

ω

 a–s

μ

ds

+S g ∞



a

a+b

ab

 a–t

ν–

(f ◦h)(t)dt

=f

ab a+b

g ∞ εγ,δ,k

 a

+εγ,δ,k

 b

+S g ∞(J+J). (.)

Settingt=_ab(u_a+–_ub) and using the harmonic convexity of|f|, we have

J=

a+b

ab



b

t– b

ν–

(f◦h)(t)dt=

a+b

ab



b

t– b

ν–

f

 t

dt

= 

b–a ab

ν  

 – u

ν–

f

ab u a+

–u  b

du

≤ 

b–a ab

ν  

 – u

ν– u f(b)+

 –u  f(a)

du.

This implies

J≤  

b–a ab

ν

ν+ 

ν(ν+ )f(b)– 

ν+ f(a)

. (.)

Similarly, using the substitutiont=_ab(v_b+–_va), we get

J=



a

a+b

ab

 a–t

ν–

(f ◦h)(t)dt=



a

a+b

ab

 a–t

ν–

f

 t

dt

≤  

b–a ab

ν

ν+ 

ν(ν+ )f(a)– 

ν+ f(b)

. (.)

Substituting (.) and (.) into (.), we obtain (.). The proof is completed.

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[a,b],g: [a,b]→Ris continuous and harmonically symmetric with respect to _aab₊_b,then the following inequality holds:

f

ab a+b ε

γ,δ,k

 a

+εγ,δ,k

 b

– εγ,δ,k

 a

+εγ,δ,k

 b

≤f

ab a+b

g ∞ εγ,δ,k

μ,ν,l,ω,(a_+_abb)+

 a

+εγ,δ,k

μ,ν,l,ω,(a_+_abb)–

 b

+S g ∞



ν

–/q 

ν+ 

/q  

ν+/q

× 

νf(b)

q

+f(a)q

/q +



νf(a)

q

+f(b)q

/q

, (.)

where h(x) =_x,x∈[_b,_a],ω= (_bab_–_a)μ_ω_,_and_|Eγ,δ,k

μ,ν,l(t)| ≤S.

Proof From (.) we have

J≤f

ab a+b

g ∞

ab a+b



b

s– b

ν– Eγ_μ,_,_νδ,_,k_l

ω

s– b

μ

ds

+S g ∞

a+b

ab



b

t– b

ν–

(f◦h)(t)dt

+f

ab a+b

g ∞



a

a+b

ab

 a–s

ν– E_μγ_,,δ_ν,_,k_l

ω

 a–s

μ

ds

+S g ∞



a

a+b

ab

 a–t

ν–

(f ◦h)(t)dt

=f

ab a+b

g ∞ εγ,δ,k

 a

+εγ,δ,k

 b

+S g ∞

a+b

ab



b

t– b ν– f  t dt+  a

a+b

ab

 a–t

ν– f  t dt .

Using the substitutionst=  ab(

u a+

–u

 b) andt=  ab(

v b+

–v

 a), the power mean inequality, and the harmonicity of|f|q_{, it follows that}

J≤f

ab a+b

g ∞ εγ,δ,k

 a

+εγ,δ,k

 b

+S g ∞ 

b–a ab ν ×    – u ν– f ab u a+

–u  b du+    – v ν– f ab v b+

–v  a

dv

≤f

ab a+b

g ∞ εγ,δ,k

 a

+εγ,δ,k

 b

(12)

+S g ∞ 

b–a ab

ν  

 – u

ν– du

–/q

×

 

 – u

ν– u f(b)

q + –u

 f(a) q

du

/q

+

 

 – v

ν– dv

–/q  

 –  v

ν– v f(a)

q + –v

 f(b) q

dv

/q .

Making the substitutionsx=  –

uandy=  – 

vand simple calculations, we get

inequal-ity (.).

4 Conclusions

We have obtained several new refinements of Hermite-Hadamard-type inequalities via harmonic convex functions. These results involve particularly the generalized Mittag-Leffler function. We also discussed several particular cases. We expect that the results obtained in this paper may stimulate further research in the field. We would like to spec-ify here that the generalized fractional integral operators containing the Mittag-Leffler function generalize the integrals of Riemann-Liouville type, and the results obtained in this paper can be developed via other different types of convexities.

Acknowledgements

The authors are thankful to the editor and anonymous referees for their helpful comments and suggestions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department Scientiﬁc-Methodical Sessions, Romanian Mathematical Society-branch Bucharest, Academy Street no. 14,}

Bucharest, RO-010014, Romania.2_{Department of Mathematics, GC University, Faisalabad, Pakistan.}3_{Department of}

Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan.

Publisher’s Note

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

Received: 1 August 2017 Accepted: 12 October 2017 References

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