Certain inequalities involving the k-struve function

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1 Nisar et al. Journal of Inequalities and Applications 7) 7:7 DOI.86/s R E S E A R C H Open Access Certain inequalities involving the -Struve function Kottaaran Sooppy Nisar, Saiful Rahman Mondal and Junesang Choi 3* * Correspondence: junesang@mail.donggu.ac.r 3 Department of Mathematics, Donggu University, Gyeongju, 3866, Republic of Korea Full list of author information is available at the end of the article Abstract We aim to introduce a -Struve function and investigate its various properties, including mainly certain inequalities associated with this function. One of the inequalities given here is pointed out to be related to the so-called classical Turán-type inequality. We also present a differential equation, several recurrence relations, and integral representations for this -Struve function. MSC: 33C; 6D7 Keywords: -Struvefunction;-gamma function; -betafunction;-digamma function; Turán-type inequalities Introduction and preliminaries Díaz and Pariguan [] introduced and investigated the so-called -gamma function Ɣ ):= t e t dt R)>; R + ). ) Here and in the following, let C, R, R +, N, andz be the sets of comple numbers, real numbers, positive real numbers, positive integers, and negative integers, respectively, and let N := N {}. For various properties of the -gamma function and its applications to generalize other related functions such as -beta function and -digamma function, we refer the interested reader, for eample, to [ 3] and the references cited therein. Nantomah and Prempeh []definedthe-digamma function := Ɣ /Ɣ whose series representation is given as follows: t):= log γ t + n= t nn + t) R + ; t C \ Z ), ) where γ is the Euler-Mascheroni constant see, e.g., [4],Section.).A calculationyields t)= n= n + t), t R + ). 3) Clearly, t)isincreasingon, ). The Authors) 7. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page of 8 Turán [5] proved that the Legendre polynomials P n ) satisfy the following determinant inequality: P n ) P n+ ) P n+ ) P n+ ) ; n N ), 4) where the equality occurs only when = ±. Recently, many researchers have applied the above classical inequality 4) in various polynomials and functions such as ultraspherical polynomials, Laguerre polynomials, Hermite polynomials, Bessel functions of the first ind, modified Bessel functions, and polygamma functions. Karlin and Szegö [6] named such determinants as in 4)Turánians. Inthispaper, weconsiderthefollowing -Struvefunction cf. [7], p. 496, Entry..3): S,c ):= c) r Ɣ r )Ɣr + 3 ) ) r+ +. 5) Then we investigate the -Struvefunction 5) asfollows: We establish certaininequalities involving S,c, one of which is shown to be related to the Turán-type inequality; we show that the -Struve function satisfies a second-order non-homogeneous differential equation; and we present an integral representation and recurrence relations for the -Struve function. Inequalities The modified -Struve function is given as L ):=S, ), 6) which is normalized and denoted by L as follows: where ) L )= Ɣ + 3 ) L )= f r, ) r+, 7) f r, ):= Ɣ + 3 ) Ɣ r )Ɣr + 3 )r+. Here, we investigate monotonicity and log-conveity involving L.Todothis,werecall some nown useful properties which are given in the following lemma see [8]). Lemma Consider the power series f )= = a and g)= = b, where a R and b R + N ). Further suppose that both series converge on < r. If the sequence {a /b } is increasing or decreasing), then the function f )/g) is also increasing or decreasing) on,r). If both f and g are even, or both are odd functions, then the above results will be applicable. Theorem Let R + be fied. Then the following statements hold.

3 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 3 of 8 i) For μ > 3/, then the function L μ )/L ) is increasing on R. ii) The function L ) is decreasing for fied [, ) and increasing for fied,). Also, the function L ) is log-conve on 3/, ) for fied R +. iii) The function L + )/L ) is decreasing on 3/, ) for fied R+. Proof To prove i), recall the series in 7). Clearly, L ) L μ ) = f r, ) r+ f rμ, ). r+ Denote w r := f r, )/f r μ, ). Then w r = Ɣ + 3 )Ɣ r + μ + 3 ) Ɣ r )Ɣ μ + 3 ). Appealing to relation 5), we find w r+ = Ɣ + 3 )Ɣ r + + μ + 3 ) Ɣ μ + 3 )Ɣ r ) w r Ɣ r )Ɣ μ + 3 ) Ɣ r + μ + 3 )Ɣ + 3 ) = r + μ + 3 )Ɣr + μ + 3 )Ɣ r ) r )Ɣr )Ɣ r + μ + 3 ) = r + μ + 3 r + + 3, whose last inequality is valid from the condition μ > 3/. Finally, the result i) follows from Lemma. For ii), since > 3, wefirstobservethecoefficientsf r, ) > for all r N.Then the logarithmic derivative of f r, )withrespectto is f r, ) f r, ) = + 3 ) r ), whose last inequality follows from ). Since f r, ) >r N ; > 3), f r, ) r N ; > 3). Hence f r, ) isdecreasingon 3/, ). This implies that, for μ > 3/, and f r, ) r+ f r μ, ) r+ ) [, ) f r, ) r+ f r μ, ) r+ ),). This proves the first statement of ii). In view of 3), we have log fr, ) )) = { n= n ) n + r ) }

4 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 4 of 8 for all R + and > 3. Therefore f r )islog-conveon 3/, ). Since a sum of log-conve functions is log-conve, the second statement of ii) is proved. For iii), it is obvious from i) that d L μ ) L ) ), 8) for all R + and μ > 3/. In view of relation 7), 8)isequivalentto μ L μ )) L )) μ L μ )) L )) 9) for all R + and μ > 3/. Considering 6)andsettingc = in7)gives d L )) = πɣ + 3 ) + L + ). ) Applying )toinequality9) andusing7), we obtain μ+ { L μ+ )L ) L + )L μ )} = μ πɣ + 3 L ) μ ) μ πɣ μ + 3 L ) ) μ+ L πɣ μ + 3 )Ɣ + 3 ) μ ) L )) ) for all R + and μ > 3/. Here, the last inequality in ) follows from the first statement of ii). Also, we find from )that L μ+ ) L μ ) L + ) L ) for all R + and μ > 3/. This proves iii). Remar One of the most significant consequences of Theorem is the Turán-type inequality for the function L. The log-conveity of L the last statement of ii) in Theorem )implies L α + α) ) L ) α) L ) α) α [, ];, R + ;, 3/, ) ). ) Choosing α = / and setting = a and = + a for some a R in ) yieldsthe following reversed Turán-type inequality cf. 4)): L ) ) L a )L +a ), R + ; a R, a 3/, )). 3)

5 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 5 of 8 3 Formulae for the -Struve function Here, we present a differential equation and recurrence relations regarding the -Struve function S,c 5). Proposition Let R + and > 3. Then the -Struve function S,c 5) satisfies the following second-order non-homogeneous differential equation: d y + dy + c ) y = 4 ) + Ɣ + )Ɣ 4) ). Proof By using the -Struve function S,c 5) and the functional relation Ɣ + )=Ɣ ), 5) we find d S,c )+ d S,c ) c) r r + ) = +) r+ + Ɣ r )Ɣr + 3 ) = c) r r + )r + +) ) r+ + Ɣ r )Ɣr + 3 ) + S,c ) = 4 c) r r + )r + + ) ) r+ + Ɣ r )Ɣr + 3 ) + S,c ) = 4 ) + Ɣ + )Ɣ ) + 4 c) r ) r+ + Ɣ r + + )Ɣr + ) + S,c ) = 4 r= ) + Ɣ + )Ɣ ) c S,c )+ S,c ). This shows that y = S,c ) the differential equation 4). Theorem Let R + and > 3. Then the following recurrence relations hold true: d S,c = S ); 6) d S,c = πɣ + 3 ) c S +,c ); 7) S ) cs +,c )= d S,c ) /) πɣ + 3 8) ); S )+cs +,c )= S,c )+ /) πɣ + 3 9) ).

6 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 6 of 8 Proof From 4)we have S,c )= c) r r+ + Ɣ r )Ɣr + 3 ) r+, + which, upon differentiating with respect to and using relation 5), yields d S,c )) = c) r r + +) r+ Ɣ r )Ɣr + 3 ) = c) r Ɣ r + + )Ɣr + 3 ) r+ + ) r+ = S,c ). This proves 6). We can establish the result 7) by a similar argument as in the proof of 6). We omit the details. Similarly,from 5), we obtain d S,c )+ S,c )= S ) ) and d S,c ) S,c )= /) πɣ + 3 ) cs +,c ). ) Adding and subtracting each side of ) and) yields, respectively, the results 8) and 9). 4 Integral representations Here, we present two integral representations for the function S,c. Theorem 3 Let R +, R)>, and α R \{}. Then and S,α )= α πɣ + ) S, α )= In particular, we have πɣ + ) ) ) t ) sin αt ) dt ) t ) sinh αt ) dt. 3) ) α cos = α π S ) 4),α and ) α cosh = α π S, α ). 5)

7 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 7 of 8 Proof We begin by recalling the -beta function see []) B, y)= Ɣ )Ɣ y) = t t) y dt Ɣ + y) R + ; min { R), Ry) } > ). 6) Replacing t by t on the right-hand-sided integral in 6), we obtain B, y)= t t ) y dt. 7) Setting =r +) and y = + / in 7)gives Ɣ r + + ) = Ɣ r +))Ɣ + ) Applying the nown identity t r t ) dt. 8) Ɣ )= Ɣ) 9) and the Legendre duplication formula see [4, 7, 9]) Ɣz)Ɣ z + ) = z πɣz) 3) to the function S,c with 8), we get S,c )= ) t ) πɣ + ) c) r ) t r+ dt. 3) r +)! Finally, setting c = ±α α R \{})in3) yields, respectively, the desired results )and 3). Further, setting = / in ) and3) yields, respectively, the desired results 4) and 5). 5 Results and discussion We introduce a -Struve function and investigate its various properties, including mainly certain inequalities associated with this function. One of the inequalities given here is pointed out to be related to the so-called classical Turán-type inequality, whose many variants have been investigated. We also present a differential equation, several recurrence relations, and integral representations forthis -Struve function. 6 Conclusions The results presented here are sure to be new and potentially useful. Since the research subject here and its related ones are competitive, the content of this paper may attract interested readers who have been interested in this and related research subjects. Competing interests The authors declare that they have no competing interests.

8 Nisar etal. Journal of Inequalities and Applications 7) 7:7 Page 8 of 8 Authors contributions The authors have contributed equally to this manuscript. They read and approved the final manuscript. Author details Department of Mathematics, College of Arts and Science-Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Wadi Ad-Dawasir, Riyadh region 99, Saudi Arabia. Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf, Al-Hasa 398, Saudi Arabia. 3 Department of Mathematics, Donggu University, Gyeongju, 3866, Republic of Korea. Acnowledgements The authors would lie to epress their deep-felt gratitude for the reviewer s detailed reviewing and useful comments. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 7 November 6 Accepted: 4 March 7 References. Díaz, R, Pariguan, E: On hypergeometric functions and -Pochhammer symbol. Divulg. Mat. 5), ). Nantomah, K, Prempeh, E: Some inequalities for the -digamma function. Math. Æterna 45), ) 3. Mubeen, S, Naz, M, Rahman, G: A note on -hypergemetric differential equations. J. Inequal. Spec. Funct. 43), ) 4. Srivastava, HM, Choi, J: Zeta and q-zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam ) 5. Turán, P: On the zeros of the polynomials of Legendre. Čas. Pěst. Math. Fys. 75, 3-95) 6. Karlin, S, Szegö, G: On certain determinants whose elements are orthogonal polynomials. J. Anal. Math. 8, ) 7. Abramowitz, M, Stegun, IA eds.): Handboo of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing. National Bureau of Standards, Applied Mathematics Series, vol. 55. Natl. Bur. of Standards, Washington 97). Reprinted by Dover Publications, New Yor, 965 see also []) 8. Biernaci, M, Krzyż, J: On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie-Słodowsa, Sect. A 9, ) 9. Andrews, GE, Asey, R, Roy, R: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 7. Cambridge University Press, Cambridge 999). Olver, FWJ, Lozier, DW, Boisvert, RF, Clar, CW eds.): NIST Handboo of Mathematical Functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington ) [with CD-ROM Windows, Macintosh and UNIX)]. Cambridge University Press, Cambridge, London and New Yor ) see also [7])