Research article Special Issues

Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions

  • Received: 22 July 2020 Accepted: 15 October 2020 Published: 11 November 2020
  • MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

  • In this paper, we first investigate some subclasses of q-starlike functions. We then apply higher-order q-derivative operators to introduce and study a new subclass of q-starlike functions, which involves the Janowski functions. Several coefficient inequalities and a sufficient condition are derived. Relevant connections with a number of earlier works on this subject are also pointed out.

    Citation: Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, H. M. Srivastava, Nazar Khan, Maslina Darus, Bilal Khan. Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions[J]. AIMS Mathematics, 2021, 6(2): 1110-1125. doi: 10.3934/math.2021067

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  • In this paper, we first investigate some subclasses of q-starlike functions. We then apply higher-order q-derivative operators to introduce and study a new subclass of q-starlike functions, which involves the Janowski functions. Several coefficient inequalities and a sufficient condition are derived. Relevant connections with a number of earlier works on this subject are also pointed out.


    The territories joining Southeast Russia, East Mongolia, and Northeast China, collectively referred to as Priamury, occupy a significant part of East Asia. Geologically, this area is a zone where the fold structures of the Central Asian (Ural-Mongol) and Pacific Orogenic Belts converge, bounded to the north and south by the Siberian (North Asian) and Sino-Korean (North China) cratons, respectively (Figure 1).

    Figure 1.  Highly productive ore clusters and districts of East Asia on a tectonic framework. Modified with additions after [4,9,10].

    In Priamury, areas where deposits of noble, nonferrous, and radioactive metals of Late Mesozoic age are concentrated display not only clustered (nodal) but also linear (belt-like) arrangements [1,2,3]. Ore-forming processes in many mineragenically specialized clusters and districts, which spread hundreds and thousands of kilometers apart, are characterized by similar evolutionary trends and relative synchronicity in formation, despite being part of different tectonic structures. [4,5,6,7]. These ore-forming processes are associated with high alkalinity magmatic formations of the Late Jurassic-Early Cretaceous age. Identifying the characteristics of the geodynamic environments in which these deposits formed is important both scientifically and practically. Furthermore, the number of the largest world-class ore clusters and districts in the worldwide is relatively small.

    My purpose of this review is to provide evidence of the influence of deep geodynamics on the significant development of ore-forming processes in certain environments and to identify the prerequisites for subsequently applying this concept in metallogenic zonation, as well as in exploratory research.

    An analysis and synthesis of geological and geophysical data on regional metallogeny in Priamury, in conjunction with geochronological and tomographic research materials on the Late Mesozoic-Cenozoic deep geodynamics in East Asia, will enable a reevaluation of the major factors that influenced the formation and location of large and super large ore clusters with Au, PGE, U, Mo, and fluorite mineralization.

    The Priamury region, located within the Asian continent and situated between the Siberian and Sino-Korean cratons, is also distinguished as the Amur plate [8]. This plate is a collage of microcontinents featuring Early Precambrian sialic crust separated by orogenic (fold-thrust) superterranes of various ages made up of transformed rock complexes from passive and active margins, marked by ophiolitic sutures [7,9,10,11]. The largest super terranes in the region, which belong to the Central Asian Orogenic Belt, include Baikal-Vitim, Selenga-Stanovoi, Mongol-Okhotsk, Solonker, and South-Mongol (Figure 1). Fragments of fold-thrust structures in the Pacific belt are represented by the Badzhal and Sikhote-Alin super terranes. Among the Priamury super terranes, the most notable are Kerulen-Argun, South Gobi, and Bureya-Jiamusi-Khanka. A distinctive feature of the region is a large Late Mesozoic igneous province, which includes layered systems of volcanic-plutonic belts (VPB) and volcanic-plutonic zones (VPZ), extensive rift depressions, syneclises, and broad fields of Cenozoic plateau basalts. Notably among the VPB are Mongolia-Priargun, Greater- and Lesser Xingan, and East-Sikhote-Alin; among the VPZ are Badzhal, Umlekan-Ogodzha, Lower Zeya, and Ichun-Yuquan. The marginal continental VPBs and their segments include East-Sikhote-Alin, Uda-Murgal, and Okhotsk-Chukotsk. Among the rifts and grabens, the largest are Songliao, Amur-Zeya, Sangjian-Middle-Amur, Syaolihe, Hulunur, Erlian, and Dzunbain.

    The Priamury region exhibits abnormally high heterogeneity in its crust and mantle. It prominently features rifting accompanied by the basification of the Earth's crust along the axial zones of depressions and the formation of Cenozoic areas of basaltic volcanism. The region is also characterized by increased elevated seismic activity [10,12,13,14]. A significant regional geological feature is the Xingan-Okhotsk fragment of the Indo-China-Chukotka (Main) gravity domain, which is approximately 150 km wide with a gravitational difference of 50–100 mGal and a total length of over 3000 km [10,13]. To the east of the Xingan-Okhotsk fragment of the Main gravity domain, the crust of Priamury is thinner, measuring 32–34 km, whereas to the west, the crust thickness increases by 10–12 km. In the western area, the lithosphere thickness also increases to approximately 150 km, whereas in the east, it decreases to 80 km [10].

    Another significant geological division in the region is the Vebirs zone (Verkhoyan-Birma), which is of Late Paleozoic-Early Mesozoic origin. This zone represents the virtual western boundary of East Asia, where the influence of the Pacific Mobile Belt structures ends. In Southeast Russia, the Vebirs zone is represented by the Baikal fragment, which is 400–500 km wide and includes several extended near-meridional faults and parts of Phanerozoic fold systems enclosed between them. The Baikal and Khubsugul rifts are confined to the axial part of the zone, as is the so-called prerift area [15], in which diatremes, dikes, and subvolcanic bodies composed of alkaline basalt rocks are known to exist in Mongolia, south of the Tunkin Valley. A consistent weakening of the influence of Mesozoic-Cenozoic geodynamics from east to west is recorded in the Vebirs zone. The eastern border of the zone coincides with the Patom-Zhuya and Onon-Tura deep strike-slip faults in Transbaikalia and with the East-Gobi and Dzunbain depressions in Mongolia. Near this zone, belts of Late Jurassic-Early Cretaceous magmatism are prominent: the Aldan belt of alkaline intrusions; the Nercha-Oldoi, Mongol-Priargun, and South-Gobi basalt-rhyolite belts; and systems of rift basins synchronous with them [10]. Manifestations of Mesozoic granitoid magmatism and contrasting basalt-rhyolite associations, as well as Lower Cretaceous coal-bearing depressions, completely disappear at the western boundary of the Vebirs zone.

    The concept of the geodynamic evolution of a region is based on the integral model of the active continental margin [9,16,17]. Many scientists agree that the major events in the formation of the structure of East Asia occurred in the Jurassic-Cretaceous and Cenozoic [18,19,20]. The eastern flank of the Central Asian Orogenic Belt in the Asia Pacific convergence zone was "opposed" by the most ancient part of the paleo-Pacific plate [21]. The active development of subduction and rifting processes in the zone led to the emergence of several fragments of the Asian continental margin-related volcano-plutonic belt [22.23]. On the basis of geophysical data of the Main gravity domain location, such fragments in the Late Jurassic-Early Cretaceous evolution of the region were Uda-Murgal, Umlekan-Ogodzha, and Great Xing'an VPB. There are several alternative viewpoints regarding the Great Xing'an belt. Some geologists consider it intracontinental [9,20], while others interpret it as a continental margin-related belt [11,24,25], with some differences in the interpretation of the spatial position of the paleo-subduction zone associated with the formation of the Great Xing'an VPB. The author concurs with Gordienko [26], who reported that the subduction zone near the VPB is likely buried under the Songliao syneclise of rifting origin. To a certain extent, this is confirmed by the presence of local mantle and asthenosphere uplifts, seismic activity, and elevated heat flow. The thinned lithosphere of the syneclise resembles that of the riftogenic trough along the coasts of the Okhotsk and Japanese margin seas [10]. With such an interpretation, the Late Mesozoic volcanic zones of Eastern Transbaikalia, and perhaps the entire Mongol-Priargun belt, are external peripheral fragments of the large Upper Amur VPB, of which the Great Xing'an belt was its internal (axial) part. This interpretation of geological and geophysical materials completely agrees with the results of many geochronological [27,28,29], petrological, geochemical [30,31], and metallogenic studies [20,32,33] and relatively simply explains the reason for the convergence of the Argun-Gonzha and Selenga-Stanovoi composite terranes in the Early Jurassic, followed by the subsequent "die-off" (closure) of the Transbaikal and Upper Amur segments of the Mongol-Okhotsk oceanic basin. It is also possible that the convergence ended at the beginning of the Middle Jurassic during the collision of the Aldan-Stanovoi and Amur plates [34]. Judging from numerous isotopic age determinations of magmatites distributed in different parts of the Upper Amur VPB, its active development terminated by the end of the Early Cretaceous or somewhat later [35,36].

    In the Late Cretaceous (100–75 Ma) along the eastern margin of Asia (from Southern China to the Siberian Craton), a western fragment of the Pacific plate, represented by the Izanagi Plate, predominantly underwent frontal subduction. This subduction process contributed to the formation of the East Sikhote-Alin arc and other magmatic arcs of the VPB, as well as the development of forearc (West Sakhalin and others) and backarc (Sanjiang-Middle Amur, Alchan, Lower Amur, etc.) rift troughs filled with volcanic-sedimentary molasse complexes. The Pribrezhnaya gradient zone, associated with East Sikhote-Alin VPB, has approximately the same variance in gravity field anomalies as Xingan-Okhotsk.

    Subsequently, in the Maastrichtian-Eocene, following the absorption of the Izanagi Plate, another intensification of transform (or strike-slip) activity between the Eurasian and Pacific plates occurred. This led, in the Oligocene-Miocene, to new counter-movements and the emergence of the Kuril and Japanese island arcs. Intense rifting during this period, which resulted in the stretching and thinning of the crust, led to the formation of the Sea of Okhotsk and the Sea of Japan, as well as extensive fields of high-alkalinity plateau basalts (Figure 2).

    Figure 2.  The location of strategic raw materials in relation to the Late Mesozoic and Cenozoic rift-related depressions and large fields of highly alkaline plateau basalts. After [1,4,6,8,10,12,44,48] with modifications and additions.

    The information provided about the existence of Late Mesozoic-Cenozoic magmatic formations of crust-mantle origin in Southeast Russia, East Mongolia, and Northeast China highlights the need for modern analysis of tomographic study results. On the basis of these data [12,32,37,38,39,40,41,42] and paleotectonic reconstructions, the subduction processes of the Pacific plate beneath the Eurasian continent have been actively developing since the Late Mesozoic. As the Pacific megaplate fragments subsided into the mantle, they transformed within the transition zone into a stagnant heterochronic composite slab (Figure 3). The slab front, which aligns well with the western contour of the large post-riftogenic Mesozoic-Cenozoic depression distribution and extensive fields of Cenozoic basalts, is projected onto the Aldan and Olekma interfluves, including their middle and upper reaches, and extends further to Southeastern Transbaikalia, East Mongolia, and Northeast China. Considering studies of the Sakhalin-South Kuril Province [43], there is reason to believe that the WNW-oriented slab flank boundaries may have been paleotransform faults preserved beneath the continent and active during subduction processes. Notably, on the present-day surface, the width of the areas where mantle formations are mapped, i.e., in the belt of probable influence of both the frontal and flank boundaries of the slab, reaches 150–200 km. [6,44]. This finding is consistent with the transverse dimensions of similar faults established by researchers examining thermal fields in the Atlantic and southeast Pacific [45].

    Figure 3.  (a) Part of the Asia-Pacific convergence megazone. (b) Section of mantle along profile along line AB showing the stagnant slab in the mantle transit zone. (c) Distribution of seismic anomalies (100∆Vp/Vp%) at a depth of 550 km and velocity scale for longitudinal seismic waves. After [4,5,6,32,49] with modifications.

    East Asia covers an area of approximately 1 million square kilometers and is situated between the Siberian and North China platforms. It is bounded to the west by the Baikal fragment of the Vebirs zone and to the east by the coasts of the Sea of Okhotsk and the Sea of Japan. Currently, more than a dozen superlarge, world-class ore clusters of the Late Jurassic-Early Cretaceous age are known in this region. These deposits include gold deposits—Aldan, Balei (Russia), and Zhao-Ye (China); uranium deposits—Elkon, Strelzovka (Russia), and Dornot (Mongolia); placer deposits, mainly platinum metal deposits—Inagli, Konder, Feklistov, and Chad (Russia). Additionally, large Mo-porphyry deposits (Bugdaya, Shakhtama, Davenda, Zhireken, Caosiyao), Cu-Mo deposits (Kultuma), and fluorite veins (Garsonui, Kalangui, Usugli, Abagaitui, etc.) have been discovered within the same territory. Information about their ages is provided in Table 1.

    Table 1.  Formation time of highly productive OMS of various specializations over a stagnant slab in East Asia.
    Metallogenic specialization of OMS Typical ore clasters and districts Age (Ma) Dating method References
    Gold-bearing Aldan (South Yakutia, Russia) 165 – 155, 145 – 140,
    135 – 130
    K-Ar (magmatites) [54,55]
    Darasun (South Transbaikalia, Russia) 160.5 ± 0.4 Rb-Sr (granodio rite porphyry) [56]
    159.6 ± 1.5 K-Ar (beresites)
    Balei (Transbaikalia, Russia) 175 ± 6, 148 ± 6, 120 ± 5 K-Ar (metasomatites) [31,57,58]
    Daqingshan (North China Craton (NCC)) 239.8 ± 3.0 Ar-Ar (sericite)
    [33]
    Zhangjiakou (NCC) 389 ± 1; 135.5 ± 0.4 U-Pb (zircon)
    Yanshan (NCC) 199 ± 2; U-Pb (zircon)
    192 – 177 Re-Os (molybdenite)
    Zhao-Ye
    (Jiaodong Peninsula, China)
    121.0 ± 2.0 Ar-Ar (sericite)
    120.6 ± 0.9 Rb-Sr (pyrite)
    159 ± 1; 116 – 132; 149 ± 5, 129 ± 1; 117 ± 3 U-Pb (zircon)
    Platinum-bearing Inagli (South Yakutia, Russia) 145.8 ± 3.2; 142.4 ± 2.0;
    133.4 ± 1; 133 – 128;
    Ar-Ar (clinopyroxenite)
    [59,60]
    Chad (Khabarovsk district, Russia) 123 ± 6; 113 ± 6; 107 ± 6 90Pt–4He
    (isoferroplatinum)

    [61,62,63]
    Konder (Khabarovsk district, Russia) 124.9 ± 1.9 U-Pb (baddeleytte)
    125.8 ± 3.8 U-Pb (zircon)
    112 ± 7 90Pt-4He
    (isoferroplatinum)
    129 ± 6 90Pt-4He (isoferroplatinum) [64]
    Uranium-bearing Elkon (South Yakutia, Russia) 150 – 130 K-Ar(magmatites) [65,66]
    135 – 130 Rb-Sr (granodiorite porphyry)
    Streltsovka (South Transbaikalia, Russia) 178 – 154;150 – 138; U-Pb (zircon), [31,51]
    126 – 117 Rb-Sr (rhyolites, granites)
    144 ± 5; 138 ± 5; 129 ± 5 K-Ar (hydromicasite) [57]
    Dornot (East Mongolia) 172 – 168; 161 ± 7
    170 – 160; 145 – 143;
    K-Ar (hydromicasite) [31,51,67]
    139 ± 2 Rb-Sr (granites)
    138 – 135 U-Pb (zircon)
    Guyuan-Duolung (Inshan-Liaohe, China) 132.6 ± 8, 9~136.4 ± 3, 1 Rb-Sr (rhyolite) [68]
    136.2 ± 2.9;
    140.2 ± 1.6; 138.6 ± 1.4
    U-Pb (zircon) [69,70]
    Fluorite-bearing Usugli (South Transbaikalia, Russia) 120 – 110 ± 5 K-Ar (muscovite) [71]
    Kalangui (South Transbaikalia, Russia) 114 – 112
    Garsonui (South Transbaikalia, Russia) 165 ± 9 K-Ar (muscovite) [31,57]
    Abagaitui (South Transbaikalia, Russia) 135 ± 6
    Molybdenum-copper-porphyry Zhireken (Eastern
    Transbaikalia, Russia)
    161.0 ± 1.6; 157.5 ± 2.0 U-Pb (zircon) [52,72,73]
    163 ± 1 Re-Os (molybdenite)
    Shakhtama (Eastern
    Transbaikalia, Russia)
    160 – 157 Re-Os (molybdenite)
    163 – 159, 160 – 153 U-Pb (zircon)
    Bugdaya (Eastern
    Transbaikalia, Russia)
    136 ± 7 K-Ar (beresites) [57]
    Caosiyao,
    (Xinghe, Inner
    Mongolia, China)
    128.6 ± 2.4; 150.9 ± 2.2 Re-Os (molybdenite) [29]
    140.1 ± 1.7; 148.5 ± 0.9 U-Pb (zircon)

     | Show Table
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    Strategic raw material reserves in the listed deposits, depending on their association with specific ore-magmatic systems (OMS) and metallogenic specialization, exceed dozens to hundreds of tons of PGE, thousands of tons of Au, dozens to hundreds of thousands of tons of uranium, hundreds of thousands to millions of tons of Mo, and millions to tens of millions of tons of fluorite [4,20,46,47].

    The common features of the listed ore clusters and districts include their locations on the edges of cratons or cratonized terranes with crustal thicknesses of 36–38 km near large gravity gradients and tectonic mélange zones [50]. These objects are characterized by their association with Late Mesozoic (Middle-Late Jurassic-Early Cretaceous) mafic and/or salic magmatic, high alkalinity formations—derivatives of deep (crust-mantle) layered chambers (magmatogens), which act as important indicators of highly productive OMS. The affiliation of large clusters and districts with such OMS is determined by the sequential localization of magma and ore-forming process derivatives in their area and their zoned position. This includes, on the one hand, intrusive and subvolcanic bodies, dikes, and on the other hand, depending on specialization—Mo-porphyry or Mo-U, Li, fluorite, Au-rare metal (with Te, Bi, W), Au-U-quartz, Au-sulfide, Au-porphyry, Au-Ag, and Au-jasperoid deposits. Zonation is combined with an increase in the content of noble metals (up to extremely high levels) in later ore bodies: stockworks and linear vein bodies [4,30]. In each district of concentration, both noble metals and uranium, as well as molybdenum mineralization, show evidence that Late Jurassic-Early Cretaceous mineralization was inherited from earlier stages, as identified among the Archean greenstone, Riphean metamorphic, and Paleozoic granitoid formations [5,51,52,53]. The listed ore clusters and districts include deposits from three evolutionary series: Gold-molybdenum, rare-polymetal-uranium, and fluorine-gold-silver.

    The Aldan ore district is key to understanding the general patterns of strategic raw material deposit distributions in East Asia. It features several gold-bearing zones [54], and a significant number of uranium-bearing zones [53], alongside concentrations of molybdenum occurrences and deposits, as well as the presence of fluorite in the Elkon ore district. Notably, the platinum-bearing alluvial deposits along the Inagli River and its tributaries are also recognized [1]. The source of the platinum group minerals in the Inagli River placers is the zoned alkaline ultramafic Inagli pluton, the dunite core of which is encased by Late Mesozoic varieties high in silica and alkalis. Additionally, other zoned platinum-bearing alkaline ultramafic massifs similar to Inagli (e.g., Konder, Feklistov, Chad, etc.), featuring placers of Au and platinum group minerals, were identified to the ESE of the Aldan district in the Inagli-Konder-Feklistov magma-metallogenic belt, which stretches over 1000 kilometers (Figure 4) [1,5].

    Figure 4.  The location of the largest and other clusters of strategic raw materials in East Asia over a stagnant oceanic slab.

    There is petrological and isotope-geochemical evidence supporting the mantle origin of mafic-ultramafic complexes in the listed zoned massifs, as well as the Cr-PGE mineralization identified here [59,74,75]. The age of its formation, dated for native Pt minerals from the Konder massif (190Pt-4He method) is 112 ± 7 Ma [61]; the 190Pt-4He ages of isoferroplatinum samples of different geneses −129 ± 6 Ma [63]; and the ages of baddeleytte and zircon (U-Pb method) from the dunite core are 124, 9 ± 1, 9 and 125, 8 ± 3, 8 Ma, respectively [62]. The data presented are quite comparable to the concentrations of gold, uranium, uranium-molybdenum, molybdenum, copper-molybdenum, and fluorite mineralization in East Asia (see Table). Generally, fluorite not only is a typomorphic mineral in U, Mo-U, and Cu-Mo deposits [76] but also forms significant fluorite deposits in many ore clusters in Transbaikalia and Mongolia.

    When the seismic tomographic and minerogenic layouts of Priamury are combined, the largest ore clusters, districts, and fields of Au, PGE, U, as well as Mo and fluorite, are in the region over the front and flank boundaries of the stagnant oceanic slab (Figure 4). The emergence of highly productive OMS in this region during the Late Mesozoic was attributed to the influence of lower mantle under subduction asthenospheric fluid-energy columns, which intensified magma and ore-forming processes in the over subduction asthenosphere, lithosphere, and Earth's crust. This impact was most effective at stagnant oceanic slab boundaries located in the transit zone of the mantle, indicating that it was determined by deep geodynamics.

    According to established theories [4,20,46,47], the impact of deep geodynamics on the Earth's crust is influenced by the decompression and dehydration processes of the oceanic slab as it moves into the mantle transition zone, followed by the advection and subsequent upwelling of fluids from the heated under subduction asthenosphere into the over subduction asthenosphere. Fluid upwelling and the resulting metasomatic transformations of the lithospheric mantle led to deformation of the lithosphere, reactivation of cratonic margin parts, and the formation of magmatogens. This sequence is evident in the locations of intermediate and peripheral magma chambers: primary chambers in the lower lithosphere within the metasomatized mantle and lower crust and associated chambers in the middle and upper parts of the Earth's crust. The intensification of magmatic and ore-forming processes has led to the development of returning mantle flows near slab boundaries and the entrapment of undepleted material from the lower mantle in ascending upper mantle plumes [32]. Given the possibility of such a scenario involving the participation of lower mantle derivatives in upper mantle plumes and subsequent mantle-crustal processes, it is logical to explain the existence of large "magmatogens", the roots of which are located several hundred kilometers below the modern surface. The emergence of the magmatogene was accompanied by a concentration of previously dispersed elements, leading to the formation of highly productive systems. This is supported by geophysical [77], isotope geochemical [74], and computational experimental data [78].

    Evidence suggests the versatility of phenomena in the convergence megazone between continental and oceanic plates, accompanied by processes such as subduction, stagnation, rifting, decompression, dehydration, fluid advection, and upwelling. The emergence of return flows of lower mantle material and its mixing with upper mantle and crustal components, along with the development of a tiered system of magmatic and ore-forming chambers, explains the formation of large ore clusters and districts in East Asia (Figure 5).

    Figure 5.  Geodynamic model of large ore clusters in East Asia.

    The proposed model for the regular formation and placement of world-class ore districts in the cratonized crust of East Asia takes into account the influence of matter and energy from two asthenospheres and the lower mantle on the intensification of ore-forming processes. This model is supported by studies [5,6,30,79] on the localization of many other ore districts over a stagnant oceanic slab in Russia, Mongolia, and China.

    The author declare she have not used Artificial Intelligence (AI) tools in the creation of this article.

    The article was written in memory of geologist Professor Vadim G. Khomich. The author expresses sincere gratitude for many years of productive collaboration, in particular, the scientific idea behind this research.

    The author is not aware of any conflict of interest.



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