New Improvements of the Jensen–Mercer Inequality for Strongly Convex Functions with Applications

In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known r...

Full description

Saved in:

Bibliographic Details
Published in:	Axioms Vol. 13; no. 8; p. 553
Main Authors:	Adil Khan, Muhammad, Ivelić Bradanović, Slavica, Mahmoud, Haitham Abbas
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01.08.2024
Subjects:	Convex analysis convex and strongly convex functions Convex functions Inequalities (Mathematics) Inequality Information theory Jensen inequality Jensen–Mercer inequality Mathematical optimization strong f-divergences Pakistan
ISSN:	2075-1680, 2075-1680
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for χ-divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end.
AbstractList	In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for χ-divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end.
Audience	Academic
Author	Mahmoud, Haitham Abbas Adil Khan, Muhammad Ivelić Bradanović, Slavica
Author_xml	– sequence: 1 givenname: Muhammad orcidid: 0000-0001-5373-4663 surname: Adil Khan fullname: Adil Khan, Muhammad – sequence: 2 givenname: Slavica surname: Ivelić Bradanović fullname: Ivelić Bradanović, Slavica – sequence: 3 givenname: Haitham Abbas orcidid: 0000-0002-7873-0586 surname: Mahmoud fullname: Mahmoud, Haitham Abbas
BookMark	eNp1kc1uGyEUhVGVSE2TbLtG6topDAzMLC2raVyl7aLpGjFwcbBmwAGcxLu-Q9-wT1Ji93cRWICO7ne43PMKHYUYAKHXlFww1pO3-tHHKVNGOtK27AU6aYhsZ1R05Oif-0t0nvOa1NVT1lF2goZP8ICX0ybFe5gglIyjw-UW8AcIGcKPb98_QjKQ8DLA3VaPvuywiwl_KSmG1bjDixju4RFfboMpPoaMH3y5xfPNZvRG75UzdOz0mOH813mKvl6-u1lcza4_v18u5tczw3hTZha4pMQ0Vne8ddRyXrsUzgyN7UXDqOuIFMaQTouW973jUgJztnGMQcucY6doefC1Ua_VJvlJp52K2qu9ENNK6VS8GUExMJUXpB8I4dT2WgzCEEu1IIyZZqhebw5edTB3W8hFreM2hdq-YqSXHZVCdrXq4lC10tXUBxdL0qZuC5M3NSDnqz6vjTMpRPcE8ANgUsw5gVPGl_2UKuhHRYl6SlP9n-bfd_5gv3_3DPATiiulzw
CitedBy_id	crossref_primary_10_5269_bspm_77015
Cites_doi	10.1007/978-1-4939-1246-9_16 10.15352/bjma/1313362976 10.1186/s13662-020-03000-5 10.1007/978-3-319-78337-6 10.1007/s13398-020-00825-3 10.1007/s00010-017-0496-5 10.3390/axioms12020195 10.1186/s13660-023-02944-y 10.1016/j.automatica.2022.110349 10.3390/e25060912 10.3390/sym12010051 10.1186/s13660-021-02735-3 10.3390/e23101238 10.1007/s13370-023-01084-2 10.1186/s13660-019-2007-9 10.3390/sym11121448
ContentType	Journal Article
Copyright	COPYRIGHT 2024 MDPI AG 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Copyright_xml	– notice: COPYRIGHT 2024 MDPI AG – notice: 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
DBID	AAYXX CITATION 3V. 7SC 7TB 7XB 8AL 8FD 8FE 8FG 8FK ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO FR3 GNUQQ HCIFZ JQ2 K7- KR7 L6V L7M L~C L~D M0N M7S P62 PHGZM PHGZT PIMPY PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U DOA
DOI	10.3390/axioms13080553
DatabaseName	CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts ProQuest Central (purchase pre-March 2016) Computing Database (Alumni Edition) Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Essentials ProQuest Central ProQuest Technology Collection ProQuest One Community College ProQuest Central Engineering Research Database ProQuest Central Student SciTech Premium Collection ProQuest Computer Science Collection Computer Science Database Civil Engineering Abstracts ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Engineering Database ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Premium ProQuest One Academic (New) Publicly Available Content Database ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China Engineering collection ProQuest Central Basic DOAJ Directory of Open Access Journals
DatabaseTitle	CrossRef Publicly Available Content Database Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection Civil Engineering Abstracts ProQuest Computing Engineering Database ProQuest Central Basic ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional ProQuest One Academic UKI Edition Materials Science & Engineering Collection Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ProQuest Central (Alumni)
DatabaseTitleList	CrossRef Publicly Available Content Database
Database_xml	– sequence: 1 dbid: DOA name: Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website – sequence: 2 dbid: PIMPY name: Publicly Available Content Database url: http://search.proquest.com/publiccontent sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics
EISSN	2075-1680
ExternalDocumentID	oai_doaj_org_article_3ecf47609b0041d9a6b6c0d1a6033c2b A807376688 10_3390_axioms13080553
GeographicLocations	Pakistan
GeographicLocations_xml	– name: Pakistan
GroupedDBID	5VS 8FE 8FG AADQD AAFWJ AAYXX ABDBF ABJCF ABUWG ACUHS ADBBV AFFHD AFKRA AFPKN AFZYC ALMA_UNASSIGNED_HOLDINGS AMVHM ARAPS AZQEC BCNDV BENPR BGLVJ BPHCQ CCPQU CITATION DWQXO EAD EAP ESX GNUQQ GROUPED_DOAJ HCIFZ IAO ITC K6V K7- KQ8 L6V M7S MODMG M~E OK1 PHGZM PHGZT PIMPY PQGLB PQQKQ PROAC PTHSS QF4 QN7 3V. 7SC 7TB 7XB 8AL 8FD 8FK FR3 JQ2 KR7 L7M L~C L~D M0N P62 PKEHL PQEST PQUKI PRINS Q9U
ID	FETCH-LOGICAL-c342t-de4710c2da845f1d440096fcb2d96231f8076cc08a65499f477e3fd2f33e53ff3
IEDL.DBID	DOA
ISICitedReferencesCount	0
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001305358600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	2075-1680
IngestDate	Tue Oct 14 15:16:44 EDT 2025 Fri Jul 25 11:57:58 EDT 2025 Tue Nov 04 18:20:27 EST 2025 Sat Nov 29 07:19:12 EST 2025 Tue Nov 18 21:25:37 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	8
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c342t-de4710c2da845f1d440096fcb2d96231f8076cc08a65499f477e3fd2f33e53ff3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0001-5373-4663 0000-0002-7873-0586
OpenAccessLink	https://doaj.org/article/3ecf47609b0041d9a6b6c0d1a6033c2b
PQID	3097817678
PQPubID	2032429
ParticipantIDs	doaj_primary_oai_doaj_org_article_3ecf47609b0041d9a6b6c0d1a6033c2b proquest_journals_3097817678 gale_infotracacademiconefile_A807376688 crossref_citationtrail_10_3390_axioms13080553 crossref_primary_10_3390_axioms13080553
PublicationCentury	2000
PublicationDate	2024-08-01
PublicationDateYYYYMMDD	2024-08-01
PublicationDate_xml	– month: 08 year: 2024 text: 2024-08-01 day: 01
PublicationDecade	2020
PublicationPlace	Basel
PublicationPlace_xml	– name: Basel
PublicationTitle	Axioms
PublicationYear	2024
Publisher	MDPI AG
Publisher_xml	– name: MDPI AG
References	Ullah (ref_19) 2019; 11 (ref_14) 2021; 24 ref_36 Sayyari (ref_11) 2023; 34 ref_35 ref_34 Dragomir (ref_17) 2002; 5 Song (ref_20) 2018; 8 Crooks (ref_33) 2021; 9 (ref_4) 2024; 2 Dragomir (ref_25) 1994; 23 Moradi (ref_7) 2018; 92 Husain (ref_12) 2020; 2020 Zhao (ref_27) 2020; 114 (ref_30) 2019; 6 (ref_15) 2011; 5 Ullah (ref_18) 2019; 13 Polyak (ref_2) 1966; 7 Fitzsimmons (ref_24) 2022; 142 Mercer (ref_8) 2003; 4 ref_23 ref_22 Mercer (ref_31) 2002; 3 ref_1 ref_3 ref_29 ref_28 Ullah (ref_21) 2019; 113 (ref_32) 1967; 2 Butt (ref_13) 2022; 2022 (ref_10) 2012; 6 ref_5 Khan (ref_16) 2022; 52 Jarad (ref_9) 2023; 2023 Chen (ref_26) 2020; 2020 ref_6
References_xml	– ident: ref_5 doi: 10.1007/978-1-4939-1246-9_16 – volume: 52 start-page: 1693 year: 2022 ident: ref_16 article-title: Mercer type variants of the Jensen–Steffensen inequality publication-title: Rocky Mt. Math. – volume: 8 start-page: 6595921 year: 2018 ident: ref_20 article-title: Integral inequalities for strongly convex functions publication-title: J. Funct. – volume: 2020 start-page: 8928691 year: 2020 ident: ref_12 article-title: New Estimates for Csiszár Divergence and Zipf–Mandelbrot Entropy via Jensen–Mercer’s Inequality publication-title: Complexity – ident: ref_3 – volume: 5 start-page: 19 year: 2011 ident: ref_15 article-title: On a Jensen-Mercer operator inequality publication-title: Banach J. Math. Anal. doi: 10.15352/bjma/1313362976 – volume: 2020 start-page: 543 year: 2020 ident: ref_26 article-title: New fractional approaches for n-polynomial P-convexity with applications in special function theory publication-title: Adv. Differ. Equ. doi: 10.1186/s13662-020-03000-5 – ident: ref_1 doi: 10.1007/978-3-319-78337-6 – volume: 114 start-page: 96 year: 2020 ident: ref_27 article-title: Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM doi: 10.1007/s13398-020-00825-3 – volume: 5 start-page: 4 year: 2002 ident: ref_17 article-title: A Refinement of Jensen’s discrete inequality for differentiable convex functions publication-title: Rgmia Res. Rep. – volume: 92 start-page: 25 year: 2018 ident: ref_7 article-title: Around Jensen’s inequality for strongly convex functions publication-title: Aequat. Math. doi: 10.1007/s00010-017-0496-5 – volume: 113 start-page: 2235 year: 2019 ident: ref_21 article-title: The concept of coordinate strongly convex functions and related inequalities publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. AMat. RACSAM – ident: ref_22 doi: 10.3390/axioms12020195 – ident: ref_35 – volume: 2023 start-page: 51 year: 2023 ident: ref_9 article-title: New stochastic fractional integral and related inequalities of Jensen–Mercer and Hermite–Hadamard–Mercer type for convex stochastic processes publication-title: J. Inequal. Appl. doi: 10.1186/s13660-023-02944-y – volume: 142 start-page: 110349 year: 2022 ident: ref_24 article-title: A note on the equivalence of a strongly convex function and its induced contractive differential equation publication-title: Automatica doi: 10.1016/j.automatica.2022.110349 – volume: 6 start-page: 125 year: 2012 ident: ref_10 article-title: On some properties of Jensen-Mercer’s functional publication-title: J. Math. Inequal. – volume: 23 start-page: 71 year: 1994 ident: ref_25 article-title: Some Converse of Jensen’s inequality and applications publication-title: Anal. Num. Theor. Approx. – volume: 4 start-page: 1 year: 2003 ident: ref_8 article-title: A variant of Jensen’s inequality publication-title: J. Ineq. Pure Appl. Math. – ident: ref_34 doi: 10.3390/e25060912 – ident: ref_6 – volume: 2 start-page: 1 year: 2024 ident: ref_4 article-title: Improvements of Jensen’s inequality and its converse for strongly convex functions with applications to strongly f-divergences publication-title: J. Math. Anal. Appl. – ident: ref_29 doi: 10.3390/sym12010051 – volume: 2 start-page: 299 year: 1967 ident: ref_32 article-title: Information-type measures of difference of probability functions and indirect observations publication-title: Studia Sci. Math. Hungar – volume: 9 start-page: 1 year: 2021 ident: ref_33 article-title: On measures of entropy and information publication-title: Tech. Note – volume: 11 start-page: 9487823 year: 2019 ident: ref_19 article-title: Integral majorization type inequalities for the functions in the sense of strongly convexity publication-title: J. Funct. Spaces – volume: 6 start-page: 2680 year: 2019 ident: ref_30 article-title: Sherman’s inequality and its converse for strongly convex functions with applications to generalized f-divergences publication-title: Turk. J. Math. – ident: ref_36 – volume: 2022 start-page: 1 year: 2022 ident: ref_13 article-title: Generalized fractal Jensen and Jensen–Mercer inequalities for harmonic convex function with applications publication-title: J. Inequal. Appl. doi: 10.1186/s13660-021-02735-3 – ident: ref_23 doi: 10.3390/e23101238 – volume: 34 start-page: 38 year: 2023 ident: ref_11 article-title: Jensen-Mercer inequality for uniformly convex functions with some applications publication-title: Afr. Mat. doi: 10.1007/s13370-023-01084-2 – volume: 13 start-page: 58 year: 2019 ident: ref_18 article-title: Majorization Theorems for strongly convex functions publication-title: J. Inequal. Appl. doi: 10.1186/s13660-019-2007-9 – volume: 24 start-page: 1093 year: 2021 ident: ref_14 article-title: Some notes on Jensen-Mercer’s type inequalities; extensions and refinements with applications publication-title: Math. Inequal. Appl. – volume: 3 start-page: 3 year: 2002 ident: ref_31 article-title: Mcd. A monotonicity property of power means publication-title: J. Ineq. Pure Appl. Math. – volume: 7 start-page: 72 year: 1966 ident: ref_2 article-title: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions publication-title: Sov. Math. Dokl. – ident: ref_28 doi: 10.3390/sym11121448
SSID	ssj0000913813
Score	2.263581
Snippet	In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality....
SourceID	doaj proquest gale crossref
SourceType	Open Website Aggregation Database Enrichment Source Index Database
StartPage	553
SubjectTerms	Convex analysis convex and strongly convex functions Convex functions Inequalities (Mathematics) Inequality Information theory Jensen inequality Jensen–Mercer inequality Mathematical optimization strong f-divergences
SummonAdditionalLinks	– databaseName: Engineering Database dbid: M7S link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1Nb9QwELWgcIBD-RZbCvIBiZNVO3Zs51QtFStAtEIqSL1Z_qwqtZuyWVB74z_wD_tLmEmyCz2UC9fYUpx4_GbmefxMyOtGoG5WFiwkb5iqkmW-4olJW_u64Z6H2KvrfzIHB_boqPk8Em7dWFa5wsQeqFMbkSPfkXjgQBjA1t3zbwxvjcLd1fEKjdvkDqokiL5073DNsaDmpRVy0GqUkN3v-IuT9qwD3La8ruU1X9RL9t8EzL23mT3433E-JJtjnEmng2E8Irfy_DG5v78Wae2ekAAARwdSoecIO9oWCu30Y0bC-ernr32selnQD_M8nL28pBDi0kNkz49PL-keVqxf0Bm4xt56KZK6dPrXlvhT8nX27sveezZeucCiVNWSpQzOiscqeavqIpJSmOOUGKrUQKAkiuVGx8it15hYFmVMliVVRcpcy1LkM7Ixb-f5OaE2G3CONuWCAjo6B60EJE-BByOj0WpC2OrnuzjqkeO1GKcO8hKcLHd9sibkzbr_-aDEcWPPtziX616ooN0_aBfHblyQTuYIo9ccNSGVSI3XQUeehNdcylgFeB1agsN1DsOKfjyuAB-HilluCn8CwFlbOyHbK0twIwB07o8ZbP27-QW5V0GcNNQUbpON5eJ7fknuxh_Lk27xqrfn3zfaATc priority: 102 providerName: ProQuest
Title	New Improvements of the Jensen–Mercer Inequality for Strongly Convex Functions with Applications
URI	https://www.proquest.com/docview/3097817678 https://doaj.org/article/3ecf47609b0041d9a6b6c0d1a6033c2b
Volume	13
WOSCitedRecordID	wos001305358600001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAON databaseName: Directory of Open Access Journals customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: DOA dateStart: 20120101 isFulltext: true titleUrlDefault: https://www.doaj.org/ providerName: Directory of Open Access Journals – providerCode: PRVHPJ databaseName: ROAD: Directory of Open Access Scholarly Resources customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: M~E dateStart: 20120101 isFulltext: true titleUrlDefault: https://road.issn.org providerName: ISSN International Centre – providerCode: PRVPQU databaseName: Computer Science Database ProQuest customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: K7- dateStart: 20120301 isFulltext: true titleUrlDefault: http://search.proquest.com/compscijour providerName: ProQuest – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: M7S dateStart: 20120301 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: BENPR dateStart: 20120301 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVPQU databaseName: Publicly Available Content Database customDbUrl: eissn: 2075-1680 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000913813 issn: 2075-1680 databaseCode: PIMPY dateStart: 20120301 isFulltext: true titleUrlDefault: http://search.proquest.com/publiccontent providerName: ProQuest
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV3NbhMxELZQ4QCHil-RUiIfkDitaq-9tveYVokobaKIglROln9RpZKgbEDtBfEOvCFPwszuNkoPFRcuPqwtrXc8nm9mdvyZkDc1R96sxAsfnS5kGU3hShYLYSpX1cwxH1p2_VM9m5nz83q-ddUX1oR19MCd4A5ECllqxZC7T_JYO-VVYJE7xYQIpUfry3S9FUy1NrjmAEWiY2kUENcfuKuL5dcGLLZhVSVuoVBL1n-XSW5xZvKY7PYOIh11E3tC7qXFU_JoumFXbZ4RD5aJdtmANrnX0GWm0E_fJ8wU__n1e4rlKit6vEjdoclrCr4pPcO095fLa3qEpeZXdAKY1qodxWwsHW39y35OPk3GH4_eFf1dCUUQslwXMQHKsFBGZ2SVeZQSg5McfBlr8HB4NkyrEJhxCiNCEKlOIscyC5EqkbN4QXYWy0V6SahJGlDNxJSR-UYlrySHqMczr0XQSg5IcSM7G3oicbzP4tJCQIGytrdlPSBvN-O_dRQad448xKXYjELq6_YBKITtFcL-SyHgdbiQFjcoTCu4_pwBfBxSXdkRSAKsqjJmQPZv1tr2O7exAg-2cA0Yvvc_ZvOKPCzBDepKBvfJznr1Pb0mD8KP9UWzGpL7h-PZ_MOwVV5oT3QxxOrTM2x_jqF_fjydf_4Lcj_5Fg
linkProvider	Directory of Open Access Journals
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3LbtQwFLVKQQIWPIsYKOAFiJVVx3YSZ4HQUBh1mIeQKFJ3xvGjqlRmymSAzo5_4D_4KL6Ee_MY6KLsumAbW4ljH9-Xfc8l5GmRIG9WSFjpbc6U8JpZwT2TOrVpwS0vXc2uP86nU31wULzbID-7XBi8VtnJxFpQ-7nDGPmOxISDJAfZ-vLkM8OqUXi62pXQaGAxCqtv4LJVL4avYX2fCTF4s7-7x9qqAsxJJZbMB5DH3AlvtUpj4pVCMz66UvgCbIEkanDtnePaZug7RZXnQUYvopQhlTFKeO8lcllJneO-GuVsHdNBjk2dyIYbUsqC79jTo_mnCvSE5mkqz-i-ukTAeYqg1m6Dm__bvNwiN1o7mvYb4N8mG2F2h1yfrEloq7ukBAFOm6BJHQOt6DxSaKdvAwbUf33_McFbPQs6nIUmt3RFwYSn7_F04PB4RXfxRv4pHYDqr3cnxaA17f915L9FPlzIT94jm7P5LNwnVIcclL_2ISJBUBbKTCXgHJa8zKXLM9UjrFts41q-dSz7cWzA70JwmLPg6JHn6_4nDdPIuT1fIXbWvZAhvH4wXxyaVuAYGRyMPuPIeakSX9iszBz3ic24lE6U8DlEnkE5BsNytk3HgJ9DRjDTh5kA5ZNp3SPbHfJMK-Aq8wd2D_7d_IRc3dufjM14OB09JNcE2ITN_cltsrlcfAmPyBX3dXlULR7Xe4mSjxcN0t-onF0j
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3LbtQwFLXKFCFY8EYMFPACxCoax3YSZ4HQ0DJiaGc0EiCVlev4UVUqkzIZoLPjH_gbPocv4d48Broouy7Yxlbi2Mf35etzCXmax8ib5eOocCaLJHcqMpy5SKjEJDkzrLA1u_5eNp2q_f18tkF-dndhMK2yk4m1oHalxRj5QOCFgzgD2ToIbVrEbGf08uRzhBWk8KS1K6fRQGTXr76B-1a9GO_AWj_jfPT6_fabqK0wEFkh-TJyHmQzs9wZJZMQOynRpA-24C4HuyAOCtx8a5kyKfpRQWaZF8HxIIRPRAgC3nuJbIJJLnmPbM7Gk9nHdYQHGTdVLBqmSCFyNjCnR-WnCrSGYkkizmjCumDAeWqh1nWjG__zLN0k11sLmw6bLXGLbPj5bXJtsqanre6QAkQ7bcIpdXS0omWg0E7fegy1__r-Y4L5Pgs6nvvm1umKgnFP3-G5weHxim5jrv4pHYFRUO9biuFsOvwrGeAu-XAhP3mP9Obl3N8nVPkMzALlfEDqoNQXqYzBbSxYkQmbpbJPom7htW2Z2LEgyLEGjwyBos8CpU-er_ufNBwk5_Z8hTha90Lu8PpBuTjUrSjSwlsYfcqQDVPGLjdpkVrmYpMyISwv4HOIQo0SDoZlTXtRA34OucL0EGYC1FKqVJ9sdSjUreir9B8IPvh38xNyBbCp98bT3YfkKgdjsUms3CK95eKLf0Qu26_Lo2rxuN1YlBxcNEp_A14mZ6Q
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=New+Improvements+of+the+Jensen%E2%80%93Mercer+Inequality+for+Strongly+Convex+Functions+with+Applications&rft.jtitle=Axioms&rft.au=Muhammad+Adil+Khan&rft.au=Slavica+Iveli%C4%87+Bradanovi%C4%87&rft.au=Haitham+Abbas+Mahmoud&rft.date=2024-08-01&rft.pub=MDPI+AG&rft.eissn=2075-1680&rft.volume=13&rft.issue=8&rft.spage=553&rft_id=info:doi/10.3390%2Faxioms13080553&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_3ecf47609b0041d9a6b6c0d1a6033c2b
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2075-1680&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2075-1680&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2075-1680&client=summon