A is a group of the formula

m is the number 1 or 2,

R^8a is in each case independently of one another a hydrogen atom or a C_1-3-alkyl, hydroxy, hydroxy-C_1-3-alkyl, amino, C_1-3-alkylamino, di-(C_1-3-alkyl)-amino, amino-C_1-3-alkyl, C_1-3-alkylamino-C_1-3-alkyl, or di-(C_1-3-alkyl)-amino-C_1-3-alkyl group, where in the aforementioned substituted 5- to 7-membered groups A, the heteroatoms O or N which may optionally be introduced as substituents with R^8a, are not separated by exactly one carbon atom from a heteroatom from the group of N, O, S,

R^8b is in each case independently of one another a hydrogen atom or a C_1-3-alkyl group,

X¹ is a carbony1, C═NR^8c, C═N—OR^8c, C═N—CN or sulphonyl group,

R^8c is in each case independently of one another a hydrogen atom, a C_1-3-alkyl, C_1-3-alkylcarbonyl, or a C_1-4-alkyloxycarbonyl group,

X² is an oxygen atom or an —NR^8b— group, where R^8b is as defined above,

X³ is a carbonyl, C═NR^8c, C═N—OR^8c, C═N—CN or sulphonyl group, where R^8c is as defined above,

X⁴ is an oxygen atom or an —NR^8c- group, where R^8c is as defined above,

Ar is a phenyl or pyridyl group,

R¹ is a hydrogen, fluorine, chlorine, bromine or iodine atom, a methyl or a methoxy group, where the hydrogen atoms of the methyl or methoxy group may optionally be wholly or partly replaced by fluorine atoms,

R² is a hydrogen or fluorine atom or a methyl group,

R³ is a hydrogen atom,

R⁴ is a C_2-4-alkenyl or C_2-4-alkynyl group,

a straight-chain or branched C_1-6-alkyl group,

where the hydrogen atoms of the straight-chain or branched

C_1-6-alkyl group may optionally be replaced wholly or partly by fluorine atoms, and which may optionally be substituted by a

C_3-5-cycloalkyl group, a nitrile, hydroxy, a C_1-5-alkyloxy group, where the hydrogen atoms of the C_1-5-alkyloxy group may optionally be replaced wholly or partly by fluorine atoms, or C_1-5-alkylcarbonyloxy, C_1-5-alkyloxycarbonyloxy, C_1-5-alkylsulphanyl, C_1-5-alkylsulphinyl, C_1-5-alkylsulphonyl, carboxy, C_1-5-alkyloxycarbonyl, aminocarbonyl, C_1-5-alkylaminocarbonyl, di-(C_1-5-alkyl)-aminocarbonyl, C_3-6-cycloalkyleniminocarbonyl, aminosulphonyl, C_1-5-alkylaminosulphonyl, di-(C_1-5-alkyl)-aminosulphonyl, C_3-6-cycloalkyleniminosulphonyl, di-(C_1-5-alkyl)-phosphoryl, amino, C_1-5-alkylamino, di-(C_1-5-alkyl)-amino, C_1-5-alkylcarbonylamino, C_1-5-alkyl-sulphonylamino, or an N—(C_1-5-alkylsulphonyl)-C_1-5-alkylamino group,

a phenyl, heteroaryl, phenyl-C_1-3-alkyl or heteroaryl-C_1-3-alkyl group,

which may optionally be substituted in the phenyl or heteroaryl moiety once to three times by identical or different substituents selected from the group consisting of halogen atoms, C_1-3-alkyl, di-(C_1-3-alkyl)-amino, hydroxy, C_1-3-alkyloxy, mono-, di - and trifluoromethoxy groups,

R⁵ is a hydrogen atom or a straight-chain or branched C_1-4-alkyl group, where the hydrogen atoms of the straight-chain or branched C_1-4-alkyl group may optionally be wholly or partly replaced by fluorine atoms, or

R⁴ and R⁵ together with the carbon atom to which they are bonded form a C_3-8-cycloalkyl group,

where one of the methylene groups of a C_4-8-cycloalkyl group may be replaced by an oxygen or sulphur atom or a sulphonyl or

—N(R^8c)— group, where R^8c is as defined above, and/or

two directly adjacent methylene groups of a C_4-8-cycloalkyl group may together be replaced by a —C(O)N(R^8b)— or —S(O)₂N(R^8b)— group, where R^8b is as defined above, and/or

three directly adjacent methylene groups of a C_6-8-cycloalkyl group may together be replaced by a —OC(O)O—, —OC(O)N(R^8b)—, —N(R^8b)C(O)N(R^8b)— or —N(R^8b)S(O)₂N(R )— group, where R^8b is as defined above,

where 1 to 2 carbon atoms of a C_3-8-cycloalkyl group may optionally be substituted independently of one another by a C_1-3-alkyl, hydroxy, C_1-3-alkyloxy, C_1-3-alkyloxycarbonyl, C_1-3-alkylamino, di-(C_1-3-alkyl)-amino group,

with the proviso that such a C_3-8-cycloalkyl group formed together from R⁴ and R⁵

in which two heteroatoms in the ring selected from the group of oxygen and nitrogen are separated from one another by exactly one optionally substituted —CH₂— group, and/or

in which one or both methylene groups in the ring which are connected directly to the carbon atom to which the groups R⁴ and R⁵ are attached are replaced by a heteroatom from the group of oxygen, nitrogen and sulphur, and/or

in which a substituent which is linked to the cyclic group, and which is distinguished by a heteroatom from the group of oxygen, nitrogen, sulphur and halogen atom being directly linked to the cyclic group, is separated from another heteroatom from the group of oxygen, nitrogen and sulphur, with the exception of the sulphone group, by exactly one optionally substituted methylene group, and/or

in which two oxygen atoms are directly connected together, is excluded,

where, unless otherwise mentioned, the term “heteroaryl group” mentioned above in the definitions means a monocyclic 5- or 6-membered heteroaryl group, where

the 6-membered heteroaryl group comprises one, two or three nitrogen atoms and

the 5-membered heteroaryl group comprises an imino group which is optionally substituted by a C_1-3-alkyl, phenyl or phenyl-C_1-3-alkyl group, or an oxygen or sulphur atom or

an imino group which is optionally substituted by a C_1-3-alkyl, phenyl, amino-C_2-3-alkyl, C_1-3-alkylamino-C_2-3-alkyl, di-(C_1-3-alkyl)-amino-C_2-3-alkyl, a C_3-6-cycloalkylenimino-C_1-3-alkyl or phenyl-C_1-3-alkyl group, or an oxygen or sulphur atom and additionally a nitrogen atom or

an imino group which is optionally substituted by a C_1-3-alkyl or phenyl-C_1-3-alkyl group, and two or three nitrogen atoms,

and additionally a phenyl ring which is optionally substituted by a fluorine, chlorine or bromine atom, a C_1-3-alkyl, hydroxy, C_1-3-alkyloxy group, amino, C_1-3-alkylamino, di-(C_1-3-alkyl)-amino or C_3-6-cycloalkylenimino group may be fused to the aforementioned monocyclic heteroaryl groups via two adjacent carbon atoms,

and the linkage takes place via a nitrogen atom or via a carbon atom of the heterocyclic moiety or of a fused phenyl ring,

where, unless otherwise mentioned, the term “halogen atom” mentioned above in the definitions means an atom from the group of fluorine, chlorine, bromine and iodine,

where the alkyl, alkynyl and alkoxy groups which are present in the aforementioned definitions and which have more than two carbon atoms may, unless otherwise mentioned, be straight-chain or branched, and the alkyl groups in the aforementioned dialkylated groups, for example the dialkylamino groups, may be identical or different,

and where the hydrogen atoms in the methyl or ethyl groups present in the aforementioned definitions may, unless otherwise mentioned, be wholly or partly replaced by fluorine atoms,

or a tautomer or salt thereof.